Theorem List for Metamath Proof Explorer - 41201-41300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | tratrbVD 41201* |
Virtual deduction proof of tratrb 40876. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
▶ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)
∧ 𝐵 ∈ 𝐴) )
| 2:1,?: e1a 40967 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐴 )
| 3:1,?: e1a 40967 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
▶ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| 4:1,?: e1a 40967 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ 𝐵 ∈ 𝐴 )
| 5:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) )
| 6:5,?: e2 40971 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝑦 )
| 7:5,?: e2 40971 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑦 ∈ 𝐵 )
| 8:2,7,4,?: e121 40996 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑦 ∈ 𝐴 )
| 9:2,6,8,?: e122 40993 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝐴 )
| 10:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝐵 ∈ 𝑥 ▶ 𝐵 ∈ 𝑥 )
| 11:6,7,10,?: e223 40975 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝐵 ∈ 𝑥 ▶ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥) )
| 12:11: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝐵 ∈ 𝑥 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥)) )
| 13:: | ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵
∧ 𝐵 ∈ 𝑥)
| 14:12,13,?: e20 41067 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ ¬ 𝐵 ∈ 𝑥 )
| 15:: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵 ▶ 𝑥 = 𝐵 )
| 16:7,15,?: e23 41095 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵 ▶ 𝑦 ∈ 𝑥 )
| 17:6,16,?: e23 41095 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵 ▶ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) )
| 18:17: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝑥 = 𝐵 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)) )
| 19:: | ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)
| 20:18,19,?: e20 41067 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ ¬ 𝑥 = 𝐵 )
| 21:3,?: e1a 40967 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑦 ∈ 𝐴
∀𝑥 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| 22:21,9,4,?: e121 40996 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ [𝑥 / 𝑥][𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥
∨ 𝑥 = 𝑦) )
| 23:22,?: e2 40971 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ [𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| 24:4,23,?: e12 41064 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ (𝑥 ∈ 𝐵 ∨ 𝐵 ∈ 𝑥 ∨ 𝑥 = 𝐵) )
| 25:14,20,24,?: e222 40976 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) ▶ 𝑥 ∈ 𝐵 )
| 26:25: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ ((𝑥 ∈ 𝑦
∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) )
| 27:: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨
𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| 28:27,?: e0a 41112 | ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
→ ∀𝑦(Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥
∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴))
| 29:28,26: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
▶ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) )
| 30:: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑥∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| 31:30,?: e0a 41112 | ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ∀𝑥(Tr 𝐴
∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴))
| 32:31,29: | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥
∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) )
| 33:32,?: e1a 40967 | ⊢ ( (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐵 )
| qed:33: | ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵) |
|
Theorem | al2imVD 41202 |
Virtual deduction proof of al2im 1814. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ ∀𝑥(𝜑 → (𝜓 → 𝜒)) )
| 2:1,?: e1a 40967 | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)) )
| 3:: | ⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓
→ ∀𝑥𝜒))
| 4:2,3,?: e10 41034 | ⊢ ( ∀𝑥(𝜑 → (𝜓 → 𝜒))
▶ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) )
| qed:4: | ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒))
→ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))) |
|
Theorem | syl5impVD 41203 |
Virtual deduction proof of syl5imp 40852. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
1:: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ (𝜑
→ (𝜓 → 𝜒)) )
| 2:1,?: e1a 40967 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ (𝜓
→ (𝜑 → 𝜒)) )
| 3:: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → 𝜓) )
| 4:3,2,?: e21 41070 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜃 → (𝜑 → 𝜒)) )
| 5:4,?: e2 40971 | ⊢ ( (𝜑 → (𝜓 → 𝜒)) , (𝜃
→ 𝜓) ▶ (𝜑 → (𝜃 → 𝜒)) )
| 6:5: | ⊢ ( (𝜑 → (𝜓 → 𝜒)) ▶ ((𝜃
→ 𝜓) → (𝜑 → (𝜃 → 𝜒))) )
| qed:6: | ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃
→ 𝜓) → (𝜑 → (𝜃 → 𝜒))))
|
(Contributed by Alan Sare, 31-Dec-2011.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒)))) |
|
Theorem | idiVD 41204 |
Virtual deduction proof of idiALT 40817. The following user's
proof is completed by invoking mmj2's unify command and using mmj2's
StepSelector to pick all remaining steps of the Metamath proof.
h1:: | ⊢ 𝜑
| qed:1,?: e0a 41112 | ⊢ 𝜑
|
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ 𝜑 ⇒ ⊢ 𝜑 |
|
Theorem | ancomstVD 41205 |
Closed form of ancoms 461. The following user's proof is completed by
invoking mmj2's unify command and using mmj2's StepSelector to pick all
remaining steps of the Metamath proof.
1:: | ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
| qed:1,?: e0a 41112 | ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓
∧ 𝜑) → 𝜒))
|
The proof of ancomst 467 is derived automatically from it.
(Contributed by
Alan Sare, 25-Dec-2011.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓 ∧ 𝜑) → 𝜒)) |
|
Theorem | ssralv2VD 41206* |
Quantification restricted to a subclass for two quantifiers. ssralv 4036
for two quantifiers. The following User's Proof is a Virtual Deduction
proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. ssralv2 40871 is ssralv2VD 41206 without
virtual deductions and was automatically derived from ssralv2VD 41206.
1:: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) ▶ (𝐴 ⊆ 𝐵
∧ 𝐶 ⊆ 𝐷) )
| 2:: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 )
| 3:1: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) ▶ 𝐴 ⊆ 𝐵 )
| 4:3,2: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐷𝜑 )
| 5:4: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷𝜑) )
| 6:5: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐷𝜑) )
| 7:: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 )
| 8:7,6: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴 ▶ ∀𝑦 ∈ 𝐷𝜑 )
| 9:1: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) ▶ 𝐶 ⊆ 𝐷 )
| 10:9,8: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑, 𝑥 ∈ 𝐴 ▶ ∀𝑦 ∈ 𝐶𝜑 )
| 11:10: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶𝜑) )
| 12:: | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)
→ ∀𝑥(𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷))
| 13:: | ⊢ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑
→ ∀𝑥∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑)
| 14:12,13,11: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐶𝜑) )
| 15:14: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) , ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐷𝜑 ▶ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑 )
| 16:15: | ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)
▶ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑) )
| qed:16: | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)
→ (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐶𝜑))
|
(Contributed by Alan Sare, 10-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝜑)) |
|
Theorem | ordelordALTVD 41207 |
An element of an ordinal class is ordinal. Proposition 7.6 of
[TakeutiZaring] p. 36. This is an alternate proof of ordelord 6216 using
the Axiom of Regularity indirectly through dford2 9086. dford2 is a
weaker definition of ordinal number. Given the Axiom of Regularity, it
need not be assumed that E Fr 𝐴 because this is inferred by the
Axiom of Regularity. The following User's Proof is a Virtual Deduction
proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. ordelordALT 40877 is ordelordALTVD 41207
without virtual deductions and was automatically derived from
ordelordALTVD 41207 using the tools program
translate..without..overwriting.cmd and the Metamath program "MM-PA>
MINIMIZE_WITH *" command.
1:: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ (Ord 𝐴
∧ 𝐵 ∈ 𝐴) )
| 2:1: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ Ord 𝐴 )
| 3:1: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ 𝐵 ∈ 𝐴 )
| 4:2: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐴 )
| 5:2: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) )
| 6:4,3: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ 𝐵 ⊆ 𝐴 )
| 7:6,6,5: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥 ∈ 𝐵
∀𝑦 ∈ 𝐵(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) )
| 8:: | ⊢ ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)
↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| 9:8: | ⊢ ∀𝑦((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)
↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| 10:9: | ⊢ ∀𝑦 ∈ 𝐴((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦
∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| 11:10: | ⊢ (∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦
∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| 12:11: | ⊢ ∀𝑥(∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦
∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| 13:12: | ⊢ ∀𝑥 ∈ 𝐴(∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
| 14:13: | ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦
∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥
∨ 𝑥 = 𝑦))
| 15:14,5: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) )
| 16:4,15,3: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ Tr 𝐵 )
| 17:16,7: | ⊢ ( (Ord 𝐴 ∧ 𝐵 ∈ 𝐴) ▶ Ord 𝐵 )
| qed:17: | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
|
(Contributed by Alan Sare, 12-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) |
|
Theorem | equncomVD 41208 |
If a class equals the union of two other classes, then it equals the union
of those two classes commuted. The following User's Proof is a Virtual
Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. equncom 4133 is equncomVD 41208 without
virtual deductions and was automatically derived from equncomVD 41208.
1:: | ⊢ ( 𝐴 = (𝐵 ∪ 𝐶) ▶ 𝐴 = (𝐵 ∪ 𝐶) )
| 2:: | ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
| 3:1,2: | ⊢ ( 𝐴 = (𝐵 ∪ 𝐶) ▶ 𝐴 = (𝐶 ∪ 𝐵) )
| 4:3: | ⊢ (𝐴 = (𝐵 ∪ 𝐶) → 𝐴 = (𝐶 ∪ 𝐵))
| 5:: | ⊢ ( 𝐴 = (𝐶 ∪ 𝐵) ▶ 𝐴 = (𝐶 ∪ 𝐵) )
| 6:5,2: | ⊢ ( 𝐴 = (𝐶 ∪ 𝐵) ▶ 𝐴 = (𝐵 ∪ 𝐶) )
| 7:6: | ⊢ (𝐴 = (𝐶 ∪ 𝐵) → 𝐴 = (𝐵 ∪ 𝐶))
| 8:4,7: | ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))
|
(Contributed by Alan Sare, 17-Feb-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵)) |
|
Theorem | equncomiVD 41209 |
Inference form of equncom 4133. The following User's Proof is a
Virtual Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. equncomi 4134 is equncomiVD 41209 without
virtual deductions and was automatically derived from equncomiVD 41209.
h1:: | ⊢ 𝐴 = (𝐵 ∪ 𝐶)
| qed:1: | ⊢ 𝐴 = (𝐶 ∪ 𝐵)
|
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ 𝐴 = (𝐵 ∪ 𝐶) ⇒ ⊢ 𝐴 = (𝐶 ∪ 𝐵) |
|
Theorem | sucidALTVD 41210 |
A set belongs to its successor. Alternate proof of sucid 6273.
The following User's Proof is a Virtual Deduction proof
completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. sucidALT 41211 is sucidALTVD 41210
without virtual deductions and was automatically derived from
sucidALTVD 41210. This proof illustrates that
completeusersproof.cmd will generate a Metamath proof from any
User's Proof which is "conventional" in the sense that no step
is a virtual deduction, provided that all necessary unification
theorems and transformation deductions are in set.mm.
completeusersproof.cmd automatically converts such a
conventional proof into a Virtual Deduction proof for which each
step happens to be a 0-virtual hypothesis virtual deduction.
The user does not need to search for reference theorem labels or
deduction labels nor does he(she) need to use theorems and
deductions which unify with reference theorems and deductions in
set.mm. All that is necessary is that each theorem or deduction
of the User's Proof unifies with some reference theorem or
deduction in set.mm or is a semantic variation of some theorem
or deduction which unifies with some reference theorem or
deduction in set.mm. The definition of "semantic variation" has
not been precisely defined. If it is obvious that a theorem or
deduction has the same meaning as another theorem or deduction,
then it is a semantic variation of the latter theorem or
deduction. For example, step 4 of the User's Proof is a
semantic variation of the definition (axiom)
suc 𝐴 = (𝐴 ∪ {𝐴}), which unifies with df-suc 6200, a
reference definition (axiom) in set.mm. Also, a theorem or
deduction is said to be a semantic variation of another
theorem or deduction if it is obvious upon cursory inspection
that it has the same meaning as a weaker form of the latter
theorem or deduction. For example, the deduction Ord 𝐴
infers ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) is a
semantic variation of the theorem (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴
∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))), which unifies with
the set.mm reference definition (axiom) dford2 9086.
h1:: | ⊢ 𝐴 ∈ V
| 2:1: | ⊢ 𝐴 ∈ {𝐴}
| 3:2: | ⊢ 𝐴 ∈ ({𝐴} ∪ 𝐴)
| 4:: | ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴)
| qed:3,4: | ⊢ 𝐴 ∈ suc 𝐴
|
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ 𝐴 ∈
V ⇒ ⊢ 𝐴 ∈ suc 𝐴 |
|
Theorem | sucidALT 41211 |
A set belongs to its successor. This proof was automatically derived
from sucidALTVD 41210 using translate_without_overwriting.cmd and
minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ 𝐴 ∈
V ⇒ ⊢ 𝐴 ∈ suc 𝐴 |
|
Theorem | sucidVD 41212 |
A set belongs to its successor. The following User's Proof is a
Virtual Deduction proof completed automatically by the tools
program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2
and Norm Megill's Metamath Proof Assistant.
sucid 6273 is sucidVD 41212 without virtual deductions and was automatically
derived from sucidVD 41212.
h1:: | ⊢ 𝐴 ∈ V
| 2:1: | ⊢ 𝐴 ∈ {𝐴}
| 3:2: | ⊢ 𝐴 ∈ (𝐴 ∪ {𝐴})
| 4:: | ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
| qed:3,4: | ⊢ 𝐴 ∈ suc 𝐴
|
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ 𝐴 ∈
V ⇒ ⊢ 𝐴 ∈ suc 𝐴 |
|
Theorem | imbi12VD 41213 |
Implication form of imbi12i 353. The following User's Proof is a Virtual
Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. imbi12 349 is imbi12VD 41213 without virtual
deductions and was automatically derived from imbi12VD 41213.
1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑 ↔ 𝜓) )
| 2:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ (𝜒 ↔ 𝜃) )
| 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜑
→ 𝜒) ▶ (𝜑 → 𝜒) )
| 4:1,3: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜑
→ 𝜒) ▶ (𝜓 → 𝜒) )
| 5:2,4: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜑
→ 𝜒) ▶ (𝜓 → 𝜃) )
| 6:5: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜑 → 𝜒) → (𝜓 → 𝜃)) )
| 7:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜓
→ 𝜃) ▶ (𝜓 → 𝜃) )
| 8:1,7: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜓
→ 𝜃) ▶ (𝜑 → 𝜃) )
| 9:2,8: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜓
→ 𝜃) ▶ (𝜑 → 𝜒) )
| 10:9: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜓 → 𝜃) → (𝜑 → 𝜒)) )
| 11:6,10: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)) )
| 12:11: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ↔ 𝜃)
→ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) )
| qed:12: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃)
→ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))))
|
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))) |
|
Theorem | imbi13VD 41214 |
Join three logical equivalences to form equivalence of implications. The
following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi13 40860
is imbi13VD 41214 without virtual deductions and was automatically derived
from imbi13VD 41214.
1:: | ⊢ ( (𝜑 ↔ 𝜓) ▶ (𝜑 ↔ 𝜓) )
| 2:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ (𝜒 ↔ 𝜃) )
| 3:: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜏
↔ 𝜂) ▶ (𝜏 ↔ 𝜂) )
| 4:2,3: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜏
↔ 𝜂) ▶ ((𝜒 → 𝜏) ↔ (𝜃 → 𝜂)) )
| 5:1,4: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃) , (𝜏
↔ 𝜂) ▶ ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂))) )
| 6:5: | ⊢ ( (𝜑 ↔ 𝜓) , (𝜒 ↔ 𝜃)
▶ ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃
→ 𝜂)))) )
| 7:6: | ⊢ ( (𝜑 ↔ 𝜓) ▶ ((𝜒 ↔ 𝜃)
→ ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃
→ 𝜂))))) )
| qed:7: | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃)
→ ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃
→ 𝜂))))))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜏 ↔ 𝜂) → ((𝜑 → (𝜒 → 𝜏)) ↔ (𝜓 → (𝜃 → 𝜂)))))) |
|
Theorem | sbcim2gVD 41215 |
Distribution of class substitution over a left-nested implication.
Similar to sbcimg 3823.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
sbcim2g 40878 is sbcim2gVD 41215 without virtual deductions and was automatically
derived from sbcim2gVD 41215.
1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| 2:: | ⊢ ( 𝐴 ∈ 𝐵 , [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ▶ [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) )
| 3:1,2: | ⊢ ( 𝐴 ∈ 𝐵 , [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ▶ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) )
| 4:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜓 → 𝜒)
↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) )
| 5:3,4: | ⊢ ( 𝐴 ∈ 𝐵 , [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ▶ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓
→ [𝐴 / 𝑥]𝜒)) )
| 6:5: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓
→ [𝐴 / 𝑥]𝜒))) )
| 7:: | ⊢ ( 𝐴 ∈ 𝐵 , ([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) ▶ ([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) )
| 8:4,7: | ⊢ ( 𝐴 ∈ 𝐵 , ([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) ▶ ([𝐴 / 𝑥]𝜑
→ [𝐴 / 𝑥](𝜓 → 𝜒)) )
| 9:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))) )
| 10:8,9: | ⊢ ( 𝐴 ∈ 𝐵 , ([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) ▶ [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) )
| 11:10: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥]𝜑
→ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒))) )
| 12:6,11: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝜑
→ (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓
→ [𝐴 / 𝑥]𝜒))) )
| qed:12: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓
→ 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓
→ [𝐴 / 𝑥]𝜒))))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))) |
|
Theorem | sbcbiVD 41216 |
Implication form of sbcbii 3832.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
sbcbi 40879 is sbcbiVD 41216 without virtual deductions and was automatically
derived from sbcbiVD 41216.
1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| 2:: | ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓)
▶ ∀𝑥(𝜑 ↔ 𝜓) )
| 3:1,2: | ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓)
▶ [𝐴 / 𝑥](𝜑 ↔ 𝜓) )
| 4:1,3: | ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓)
▶ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓) )
| 5:4: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑥(𝜑 ↔ 𝜓)
→ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) )
| qed:5: | ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓)
→ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
|
Theorem | trsbcVD 41217* |
Formula-building inference rule for class substitution, substituting a
class variable for the setvar variable of the transitivity predicate.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
trsbc 40880 is trsbcVD 41217 without virtual deductions and was automatically
derived from trsbcVD 41217.
1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| 2:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑧 ∈ 𝑦
↔ 𝑧 ∈ 𝑦) )
| 3:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝑥
↔ 𝑦 ∈ 𝐴) )
| 4:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑧 ∈ 𝑥
↔ 𝑧 ∈ 𝐴) )
| 5:1,2,3,4: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥]𝑧 ∈ 𝑦
→ ([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴))) )
| 6:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 →
([𝐴 / 𝑥]𝑦 ∈ 𝑥 → [𝐴 / 𝑥]𝑧 ∈ 𝑥))) )
| 7:5,6: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴
→ 𝑧 ∈ 𝐴))) )
| 8:: | ⊢ ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴
→ 𝑧 ∈ 𝐴)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
| 9:7,8: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| 10:: | ⊢ ((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥
→ 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
| 11:10: | ⊢ ∀𝑥((𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑥
→ 𝑧 ∈ 𝑥)) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
| 12:1,11: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦
→ (𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑥)) ↔ [𝐴 / 𝑥]((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)
→ 𝑧 ∈ 𝑥)) )
| 13:9,12: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| 14:13: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑦([𝐴 / 𝑥]((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| 15:14: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑦[𝐴 / 𝑥]((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| 16:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑦((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) )
| 17:15,16: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑦((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| 18:17: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑧([𝐴 / 𝑥]∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
→ 𝑧 ∈ 𝐴)) )
| 19:18: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑧[𝐴 / 𝑥]∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) )
| 20:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑧∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧[𝐴 / 𝑥]∀𝑦((𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) )
| 21:19,20: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑧∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) )
| 22:: | ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
| 23:21,22: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑧∀𝑦((
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ Tr 𝐴) )
| 24:: | ⊢ (Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦
∈ 𝑥) → 𝑧 ∈ 𝑥))
| 25:24: | ⊢ ∀𝑥(Tr 𝑥 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦
∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
| 26:1,25: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]Tr 𝑥
↔ [𝐴 / 𝑥]∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) )
| 27:23,26: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]Tr 𝑥
↔ Tr 𝐴) )
| qed:27: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]Tr 𝑥
↔ Tr 𝐴))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴)) |
|
Theorem | truniALTVD 41218* |
The union of a class of transitive sets is transitive.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
truniALT 40881 is truniALTVD 41218 without virtual deductions and was
automatically derived from truniALTVD 41218.
1:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ∀𝑥 ∈ 𝐴
Tr 𝑥 )
| 2:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) )
| 3:2: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ 𝑧 ∈ 𝑦 )
| 4:2: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ 𝑦 ∈ ∪ 𝐴 )
| 5:4: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) )
| 6:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) )
| 7:6: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ 𝑦 ∈ 𝑞 )
| 8:6: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ 𝑞 ∈ 𝐴 )
| 9:1,8: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ [𝑞 / 𝑥]Tr 𝑥 )
| 10:8,9: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ Tr 𝑞 )
| 11:3,7,10: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ 𝑧 ∈ 𝑞 )
| 12:11,8: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴), (𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) ▶ 𝑧 ∈ ∪ 𝐴 )
| 13:12: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| 14:13: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ ∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| 15:14: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| 16:5,15: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) ▶ 𝑧 ∈ ∪ 𝐴 )
| 17:16: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ((𝑧 ∈ 𝑦
∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| 18:17: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥
▶ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴) )
| 19:18: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ Tr ∪ 𝐴 )
| qed:19: | ⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∪ 𝐴)
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪
𝐴) |
|
Theorem | ee33VD 41219 |
Non-virtual deduction form of e33 41074.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
ee33 40861 is ee33VD 41219 without virtual deductions and was automatically
derived from ee33VD 41219.
h1:: | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
| h2:: | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
| h3:: | ⊢ (𝜃 → (𝜏 → 𝜂))
| 4:1,3: | ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂))))
| 5:4: | ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
| 6:2,5: | ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓
→ (𝜒 → 𝜂))))))
| 7:6: | ⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒
→ 𝜂)))))
| 8:7: | ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
| qed:8: | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
|
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) & ⊢ (𝜃 → (𝜏 → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) |
|
Theorem | trintALTVD 41220* |
The intersection of a class of transitive sets is transitive. Virtual
deduction proof of trintALT 41221.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
trintALT 41221 is trintALTVD 41220 without virtual deductions and was
automatically derived from trintALTVD 41220.
1:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ∀𝑥 ∈ 𝐴Tr 𝑥 )
| 2:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) )
| 3:2: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ 𝑧 ∈ 𝑦 )
| 4:2: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ 𝑦 ∈ ∩ 𝐴 )
| 5:4: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ ∀𝑞 ∈ 𝐴𝑦 ∈ 𝑞 )
| 6:5: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞) )
| 7:: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ 𝑞 ∈ 𝐴 )
| 8:7,6: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ 𝑦 ∈ 𝑞 )
| 9:7,1: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ [𝑞 / 𝑥]Tr 𝑥 )
| 10:7,9: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ Tr 𝑞 )
| 11:10,3,8: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴), 𝑞 ∈ 𝐴 ▶ 𝑧 ∈ 𝑞 )
| 12:11: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞) )
| 13:12: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ ∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞) )
| 14:13: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ ∀𝑞 ∈ 𝐴𝑧 ∈ 𝑞 )
| 15:3,14: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
∩ 𝐴) ▶ 𝑧 ∈ ∩ 𝐴 )
| 16:15: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ((𝑧 ∈ 𝑦 ∧ 𝑦
∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴) )
| 17:16: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ ∀𝑧∀𝑦((𝑧
∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴) )
| 18:17: | ⊢ ( ∀𝑥 ∈ 𝐴Tr 𝑥 ▶ Tr ∩ 𝐴 )
| qed:18: | ⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∩ 𝐴)
|
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩
𝐴) |
|
Theorem | trintALT 41221* |
The intersection of a class of transitive sets is transitive. Exercise
5(b) of [Enderton] p. 73. trintALT 41221 is an alternate proof of trint 5191.
trintALT 41221 is trintALTVD 41220 without virtual deductions and was
automatically derived from trintALTVD 41220 using the tools program
translate..without..overwriting.cmd and the Metamath program
"MM-PA>
MINIMIZE_WITH *" command. (Contributed by Alan Sare, 17-Apr-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩
𝐴) |
|
Theorem | undif3VD 41222 |
The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual
deduction proof of undif3 4268.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
undif3 4268 is undif3VD 41222 without virtual deductions and was automatically
derived from undif3VD 41222.
1:: | ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴
∨ 𝑥 ∈ (𝐵 ∖ 𝐶)))
| 2:: | ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈
𝐶))
| 3:2: | ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥
∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| 4:1,3: | ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴
∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| 5:: | ⊢ ( 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 )
| 6:5: | ⊢ ( 𝑥 ∈ 𝐴 ▶ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) )
| 7:5: | ⊢ ( 𝑥 ∈ 𝐴 ▶ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴) )
| 8:6,7: | ⊢ ( 𝑥 ∈ 𝐴 ▶ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧
(¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)) )
| 9:8: | ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (
¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| 10:: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ (𝑥 ∈ 𝐵
∧ ¬ 𝑥 ∈ 𝐶) )
| 11:10: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ 𝑥 ∈ 𝐵 )
| 12:10: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ ¬ 𝑥 ∈ 𝐶
)
| 13:11: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ (𝑥 ∈ 𝐴
∨ 𝑥 ∈ 𝐵) )
| 14:12: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ (¬ 𝑥 ∈
𝐶 ∨ 𝑥 ∈ 𝐴) )
| 15:13,14: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ ((𝑥 ∈
𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)) )
| 16:15: | ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴
∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| 17:9,16: | ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
→ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| 18:: | ⊢ ( (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ▶ (𝑥 ∈ 𝐴
∧ ¬ 𝑥 ∈ 𝐶) )
| 19:18: | ⊢ ( (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ▶ 𝑥 ∈ 𝐴 )
| 20:18: | ⊢ ( (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ▶ ¬ 𝑥 ∈ 𝐶
)
| 21:18: | ⊢ ( (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ▶ (𝑥 ∈ 𝐴
∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) )
| 22:21: | ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∨
(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| 23:: | ⊢ ( (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ▶ (𝑥 ∈ 𝐴 ∧
𝑥 ∈ 𝐴) )
| 24:23: | ⊢ ( (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ▶ 𝑥 ∈ 𝐴 )
| 25:24: | ⊢ ( (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ▶ (𝑥 ∈ 𝐴 ∨
(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) )
| 26:25: | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∨ (
𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| 27:10: | ⊢ ( (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ▶ (𝑥 ∈ 𝐴
∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) )
| 28:27: | ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∨
(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| 29:: | ⊢ ( (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ▶ (𝑥 ∈ 𝐵 ∧
𝑥 ∈ 𝐴) )
| 30:29: | ⊢ ( (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ▶ 𝑥 ∈ 𝐴 )
| 31:30: | ⊢ ( (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ▶ (𝑥 ∈ 𝐴 ∨
(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) )
| 32:31: | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∨ (
𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| 33:22,26: | ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐴
∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| 34:28,32: | ⊢ (((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵
∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| 35:33,34: | ⊢ ((((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈
𝐴 ∧ 𝑥 ∈ 𝐴)) ∨ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)))
→ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| 36:: | ⊢ ((((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈
𝐴 ∧ 𝑥 ∈ 𝐴)) ∨ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)))
↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| 37:36,35: | ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶
∨ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
| 38:17,37: | ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| 39:: | ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈
𝐴))
| 40:39: | ⊢ (¬ 𝑥 ∈ (𝐶 ∖ 𝐴) ↔ ¬ (𝑥 ∈ 𝐶 ∧
¬ 𝑥 ∈ 𝐴))
| 41:: | ⊢ (¬ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴) ↔ (¬ 𝑥
∈ 𝐶 ∨ 𝑥 ∈ 𝐴))
| 42:40,41: | ⊢ (¬ 𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥
∈ 𝐴))
| 43:: | ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵
))
| 44:43,42: | ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ (𝐶 ∖ 𝐴)
) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴)))
| 45:: | ⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) ↔ (
𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ (𝐶 ∖ 𝐴)))
| 46:45,44: | ⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) ↔ (
(𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| 47:4,38: | ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴
∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
| 48:46,47: | ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴
∪ 𝐵) ∖ (𝐶 ∖ 𝐴)))
| 49:48: | ⊢ ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈
((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)))
| qed:49: | ⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ (𝐶
∖ 𝐴))
|
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) |
|
Theorem | sbcssgVD 41223 |
Virtual deduction proof of sbcssg 4466.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
sbcssg 4466 is sbcssgVD 41223 without virtual deductions and was automatically
derived from sbcssgVD 41223.
1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| 2:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦
∈ ⦋𝐴 / 𝑥⦌𝐶) )
| 3:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦
∈ ⦋𝐴 / 𝑥⦌𝐷) )
| 4:2,3: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥]𝑦 ∈ 𝐶 →
[𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷
)) )
| 5:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑦 ∈ 𝐶 →
𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷)) )
| 6:4,5: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑦 ∈ 𝐶 →
𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) )
| 7:6: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑦([𝐴 / 𝑥](𝑦 ∈
𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) )
| 8:7: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑦[𝐴 / 𝑥](𝑦 ∈
𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)
) )
| 9:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑦(𝑦 ∈
𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)) )
| 10:8,9: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]∀𝑦(𝑦 ∈
𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)
) )
| 11:: | ⊢ (𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))
| 110:11: | ⊢ ∀𝑥(𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈
𝐷))
| 12:1,110: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔
[𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)) )
| 13:10,12: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔
∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) )
| 14:: | ⊢ (⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ↔ ∀
𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))
| 15:13,14: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔
⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷) )
| qed:15: | ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋
𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷))
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷)) |
|
Theorem | csbingVD 41224 |
Virtual deduction proof of csbin 4394.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbin 4394 is csbingVD 41224 without virtual deductions and was
automatically derived from csbingVD 41224.
1:: | ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 )
| 2:: | ⊢ (𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)
}
| 20:2: | ⊢ ∀𝑥(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦
∈ 𝐷)}
| 30:1,20: | ⊢ ( 𝐴 ∈ 𝐵 ▶ [𝐴 / 𝑥](𝐶 ∩ 𝐷) =
{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} )
| 3:1,30: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) =
⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} )
| 4:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶
∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} )
| 5:3,4: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) =
{𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} )
| 6:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦
∈ ⦋𝐴 / 𝑥⦌𝐶) )
| 7:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦
∈ ⦋𝐴 / 𝑥⦌𝐷) )
| 8:6,7: | ⊢ ( 𝐴 ∈ 𝐵 ▶ (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧
[𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷
)) )
| 9:1: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧
𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷)) )
| 10:9,8: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧
𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) )
| 11:10: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ∀𝑦([𝐴 / 𝑥](𝑦 ∈
𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) )
| 12:11: | ⊢ ( 𝐴 ∈ 𝐵 ▶ {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶
∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)} )
| 13:5,12: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) =
{𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)} )
| 14:: | ⊢ (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷) = {
𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}
| 15:13,14: | ⊢ ( 𝐴 ∈ 𝐵 ▶ ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) =
(⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷) )
| qed:15: | ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (
⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷)) |
|
Theorem | onfrALTlem5VD 41225* |
Virtual deduction proof of onfrALTlem5 40882.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALTlem5 40882 is onfrALTlem5VD 41225 without virtual deductions and was
automatically derived from onfrALTlem5VD 41225.
1:: | ⊢ 𝑎 ∈ V
| 2:1: | ⊢ (𝑎 ∩ 𝑥) ∈ V
| 3:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔ (𝑎
∩ 𝑥) = ∅)
| 4:3: | ⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔
¬ (𝑎 ∩ 𝑥) = ∅)
| 5:: | ⊢ ((𝑎 ∩ 𝑥) ≠ ∅ ↔ ¬ (𝑎 ∩ 𝑥
) = ∅)
| 6:4,5: | ⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔
(𝑎 ∩ 𝑥) ≠ ∅)
| 7:2: | ⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔
[(𝑎 ∩ 𝑥) / 𝑏]¬ 𝑏 = ∅)
| 8:: | ⊢ (𝑏 ≠ ∅ ↔ ¬ 𝑏 = ∅)
| 9:8: | ⊢ ∀𝑏(𝑏 ≠ ∅ ↔ ¬ 𝑏 = ∅)
| 10:2,9: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅ ↔
[(𝑎 ∩ 𝑥) / 𝑏]¬ 𝑏 = ∅)
| 11:7,10: | ⊢ (¬ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 = ∅ ↔
[(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅)
| 12:6,11: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅ ↔ (
𝑎 ∩ 𝑥) ≠ ∅)
| 13:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ⊆ (𝑎 ∩ 𝑥
) ↔ (𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥))
| 14:12,13: | ⊢ (([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ⊆ (𝑎 ∩
𝑥) ∧ [(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅) ↔ ((𝑎 ∩ 𝑥) ⊆ (𝑎
∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅))
| 15:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩
𝑥) ∧ 𝑏 ≠ ∅) ↔ ([(𝑎 ∩ 𝑥) / 𝑏]𝑏 ⊆ (𝑎 ∩ 𝑥) ∧
[(𝑎 ∩ 𝑥) / 𝑏]𝑏 ≠ ∅))
| 16:15,14: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩
𝑥) ∧ 𝑏 ≠ ∅) ↔ ((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥)
≠ ∅))
| 17:2: | ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = (
⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑏 ∩ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑦)
| 18:2: | ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑏 = (𝑎 ∩ 𝑥)
| 19:2: | ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑦 = 𝑦
| 20:18,19: | ⊢ (⦋(𝑎 ∩ 𝑥) / 𝑏⦌𝑏 ∩ ⦋(𝑎
∩ 𝑥) / 𝑏⦌𝑦) = ((𝑎 ∩ 𝑥) ∩ 𝑦)
| 21:17,20: | ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = ((
𝑎 ∩ 𝑥) ∩ 𝑦)
| 22:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) =
∅ ↔ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) = ⦋(𝑎 ∩ 𝑥) / 𝑏⦌
∅)
| 23:2: | ⊢ ⦋(𝑎 ∩ 𝑥) / 𝑏⦌∅ = ∅
| 24:21,23: | ⊢ (⦋(𝑎 ∩ 𝑥) / 𝑏⦌(𝑏 ∩ 𝑦) =
⦋(𝑎 ∩ 𝑥) / 𝑏⦌∅ ↔ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)
| 25:22,24: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) =
∅ ↔ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)
| 26:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]𝑦 ∈ 𝑏 ↔ 𝑦 ∈
(𝑎 ∩ 𝑥))
| 27:25,26: | ⊢ (([(𝑎 ∩ 𝑥) / 𝑏]𝑦 ∈ 𝑏 ∧ [
(𝑎 ∩ 𝑥) / 𝑏](𝑏 ∩ 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((
𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
| 28:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏
∩ 𝑦) = ∅) ↔ ([(𝑎 ∩ 𝑥) / 𝑏]𝑦 ∈ 𝑏 ∧ [(𝑎 ∩ 𝑥)
/ 𝑏](𝑏 ∩ 𝑦) = ∅))
| 29:27,28: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏 ∧ (𝑏
∩ 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) =
∅))
| 30:29: | ⊢ ∀𝑦([(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏
∧ (𝑏 ∩ 𝑦) = ∅) ↔ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅))
| 31:30: | ⊢ (∃𝑦[(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏
∧ (𝑏 ∩ 𝑦) = ∅) ↔ ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥)
∩ 𝑦) = ∅))
| 32:: | ⊢ (∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩
𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅
))
| 33:31,32: | ⊢ (∃𝑦[(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏
∧ (𝑏 ∩ 𝑦) = ∅) ↔ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦)
= ∅)
| 34:2: | ⊢ (∃𝑦[(𝑎 ∩ 𝑥) / 𝑏](𝑦 ∈ 𝑏
∧ (𝑏 ∩ 𝑦) = ∅) ↔ [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦(𝑦 ∈ 𝑏 ∧ (
𝑏 ∩ 𝑦) = ∅))
| 35:33,34: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]∃𝑦(𝑦 ∈ 𝑏
∧ (𝑏 ∩ 𝑦) = ∅) ↔ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅)
| 36:: | ⊢ (∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅ ↔ ∃𝑦
(𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅))
| 37:36: | ⊢ ∀𝑏(∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅ ↔
∃𝑦(𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦) = ∅))
| 38:2,37: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏
∩ 𝑦) = ∅ ↔ [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦(𝑦 ∈ 𝑏 ∧ (𝑏 ∩ 𝑦)
= ∅))
| 39:35,38: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏
∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)
| 40:16,39: | ⊢ (([(𝑎 ∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎
∩ 𝑥) ∧ 𝑏 ≠ ∅) → [(𝑎 ∩ 𝑥) / 𝑏]∃𝑦 ∈ 𝑏(𝑏 ∩
𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠
∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
| 41:2: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎
∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ ([(𝑎
∩ 𝑥) / 𝑏](𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → [(𝑎 ∩ 𝑥) /
𝑏]∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅))
| qed:40,41: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎
∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎
∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥
)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢
([(𝑎 ∩
𝑥) / 𝑏]((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏 (𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)) |
|
Theorem | onfrALTlem4VD 41226* |
Virtual deduction proof of onfrALTlem4 40883.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALTlem4 40883 is onfrALTlem4VD 41226 without virtual deductions and was
automatically derived from onfrALTlem4VD 41226.
1:: | ⊢ 𝑦 ∈ V
| 2:1: | ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ ⦋
𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅)
| 3:1: | ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (⦋𝑦 / 𝑥⦌
𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥)
| 4:1: | ⊢ ⦋𝑦 / 𝑥⦌𝑎 = 𝑎
| 5:1: | ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦
| 6:4,5: | ⊢ (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) = (
𝑎 ∩ 𝑦)
| 7:3,6: | ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (𝑎 ∩ 𝑦)
| 8:1: | ⊢ ⦋𝑦 / 𝑥⦌∅ = ∅
| 9:7,8: | ⊢ (⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌
∅ ↔ (𝑎 ∩ 𝑦) = ∅)
| 10:2,9: | ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ (𝑎
∩ 𝑦) = ∅)
| 11:1: | ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎)
| 12:11,10: | ⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](
𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
| 13:1: | ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) =
∅) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅))
| qed:13,12: | ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) =
∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
|
Theorem | onfrALTlem3VD 41227* |
Virtual deduction proof of onfrALTlem3 40884.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALTlem3 40884 is onfrALTlem3VD 41227 without virtual deductions and was
automatically derived from onfrALTlem3VD 41227.
1:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ (𝑎
⊆ On ∧ 𝑎 ≠ ∅) )
| 2:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) )
| 3:2: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ 𝑥 ∈ 𝑎 )
| 4:1: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ 𝑎 ⊆
On )
| 5:3,4: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ 𝑥 ∈ On )
| 6:5: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ Ord 𝑥 )
| 7:6: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ E We 𝑥 )
| 8:: | ⊢ (𝑎 ∩ 𝑥) ⊆ 𝑥
| 9:7,8: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ E We (𝑎 ∩ 𝑥) )
| 10:9: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ E Fr (𝑎 ∩ 𝑥) )
| 11:10: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∀𝑏((𝑏 ⊆ (𝑎 ∩ 𝑥) ∧ 𝑏 ≠
∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) )
| 12:: | ⊢ 𝑥 ∈ V
| 13:12,8: | ⊢ (𝑎 ∩ 𝑥) ∈ V
| 14:13,11: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ [(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎
∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) )
| 15:: | ⊢ ([(𝑎 ∩ 𝑥) / 𝑏]((𝑏 ⊆ (𝑎
∩ 𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦 ∈ 𝑏(𝑏 ∩ 𝑦) = ∅) ↔ (((𝑎 ∩
𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)(
(𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
| 16:14,15: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ (((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (
𝑎 ∩ 𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) =
∅) )
| 17:: | ⊢ (𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥)
| 18:2: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ¬ (𝑎 ∩ 𝑥) = ∅ )
| 19:18: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ (𝑎 ∩ 𝑥) ≠ ∅ )
| 20:17,19: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ((𝑎 ∩ 𝑥) ⊆ (𝑎 ∩ 𝑥) ∧ (𝑎 ∩
𝑥) ≠ ∅) )
| qed:16,20: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅ )
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅ ) |
|
Theorem | simplbi2comtVD 41228 |
Virtual deduction proof of simplbi2comt 504.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
simplbi2comt 504 is simplbi2comtVD 41228 without virtual deductions and was
automatically derived from simplbi2comtVD 41228.
1:: | ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ (𝜑 ↔ (
𝜓 ∧ 𝜒)) )
| 2:1: | ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ ((𝜓 ∧ 𝜒
) → 𝜑) )
| 3:2: | ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ (𝜓 → (𝜒
→ 𝜑)) )
| 4:3: | ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ (𝜒 → (𝜓
→ 𝜑)) )
| qed:4: | ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓
→ 𝜑)))
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑))) |
|
Theorem | onfrALTlem2VD 41229* |
Virtual deduction proof of onfrALTlem2 40886.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALTlem2 40886 is onfrALTlem2VD 41229 without virtual deductions and was
automatically derived from onfrALTlem2VD 41229.
1:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩
𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) )
| 2:1: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑧 ∈ (𝑎 ∩ 𝑦) )
| 3:2: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑧 ∈ 𝑎 )
| 4:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ (𝑎
⊆ On ∧ 𝑎 ≠ ∅) )
| 5:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) )
| 6:5: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ 𝑥 ∈ 𝑎 )
| 7:4: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ 𝑎 ⊆
On )
| 8:6,7: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ 𝑥 ∈ On )
| 9:8: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ Ord 𝑥 )
| 10:9: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ Tr 𝑥 )
| 11:1: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑦 ∈ (𝑎 ∩ 𝑥) )
| 12:11: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑦 ∈ 𝑥 )
| 13:2: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑧 ∈ 𝑦 )
| 14:10,12,13: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑧 ∈ 𝑥 )
| 15:3,14: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑧 ∈ (𝑎 ∩ 𝑥) )
| 16:13,15: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩
𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) ▶ 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦) )
| 17:16: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ (𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)) )
| 18:17: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ ∀𝑧(𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)) )
| 19:18: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ (𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦) )
| 20:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) )
| 21:20: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅ )
| 22:19,21: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ (𝑎 ∩ 𝑦) = ∅ )
| 23:20: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ 𝑦 ∈ (𝑎 ∩ 𝑥) )
| 24:23: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ 𝑦 ∈ 𝑎 )
| 25:22,24: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅), (𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅) ▶ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅) )
| 26:25: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥)
∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) )
| 27:26: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∀𝑦((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥
) ∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) )
| 28:27: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ (∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥
) ∩ 𝑦) = ∅) → ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) )
| 29:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦
) = ∅ )
| 30:29: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥)
∩ 𝑦) = ∅) )
| 31:28,30: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅) )
| qed:31: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈
𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ )
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅ ) |
|
Theorem | onfrALTlem1VD 41230* |
Virtual deduction proof of onfrALTlem1 40888.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALTlem1 40888 is onfrALTlem1VD 41230 without virtual deductions and was
automatically derived from onfrALTlem1VD 41230.
1:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧
(𝑎 ∩ 𝑥) = ∅) ▶ (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) )
| 2:1: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧
(𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) )
| 3:2: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧
(𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)
)
| 4:: | ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅
) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
| 5:4: | ⊢ ∀𝑦([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥)
= ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
| 6:5: | ⊢ (∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥)
= ∅) ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
| 7:3,6: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧
(𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅) )
| 8:: | ⊢ (∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ ↔ ∃𝑦(
𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
| qed:7,8: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧
(𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ )
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅ ) |
|
Theorem | onfrALTVD 41231 |
Virtual deduction proof of onfrALT 40889.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
onfrALT 40889 is onfrALTVD 41231 without virtual deductions and was
automatically derived from onfrALTVD 41231.
1:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎
∧ ¬ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ )
| 2:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , (𝑥 ∈ 𝑎
∧ (𝑎 ∩ 𝑥) = ∅) ▶ ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ )
| 3:1: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , 𝑥 ∈ 𝑎 ▶
(¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) )
| 4:2: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , 𝑥 ∈ 𝑎 ▶
((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) )
| 5:: | ⊢ ((𝑎 ∩ 𝑥) = ∅ ∨ ¬ (𝑎 ∩ 𝑥) =
∅)
| 6:5,4,3: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) , 𝑥 ∈ 𝑎 ▶
∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ )
| 7:6: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ (𝑥 ∈ 𝑎
→ ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) )
| 8:7: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ ∀𝑥(𝑥
∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) )
| 9:8: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ (∃𝑥𝑥
∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) )
| 10:: | ⊢ (𝑎 ≠ ∅ ↔ ∃𝑥𝑥 ∈ 𝑎)
| 11:9,10: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ (𝑎 ≠
∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅) )
| 12:: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ (𝑎 ⊆
On ∧ 𝑎 ≠ ∅) )
| 13:12: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ 𝑎 ≠
∅ )
| 14:13,11: | ⊢ ( (𝑎 ⊆ On ∧ 𝑎 ≠ ∅) ▶ ∃𝑦 ∈
𝑎(𝑎 ∩ 𝑦) = ∅ )
| 15:14: | ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎
(𝑎 ∩ 𝑦) = ∅)
| 16:15: | ⊢ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦
∈ 𝑎(𝑎 ∩ 𝑦) = ∅)
| qed:16: | ⊢ E Fr On
|
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ E Fr
On |
|
Theorem | csbeq2gVD 41232 |
Virtual deduction proof of csbeq2 3891.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbeq2 3891 is csbeq2gVD 41232 without virtual deductions and was
automatically derived from csbeq2gVD 41232.
1:: | ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 )
| 2:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∀𝑥𝐵 = 𝐶 → [𝐴 / 𝑥]
𝐵 = 𝐶) )
| 3:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴
/ 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) )
| 4:2,3: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∀𝑥𝐵 = 𝐶 → ⦋𝐴 / 𝑥
⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) )
| qed:4: | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌
𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) |
|
Theorem | csbsngVD 41233 |
Virtual deduction proof of csbsng 4647.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbsng 4647 is csbsngVD 41233 without virtual deductions and was automatically
derived from csbsngVD 41233.
1:: | ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 )
| 2:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑦 = 𝐵
↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵) )
| 3:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌𝑦 = 𝑦 )
| 4:3: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴
/ 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵) )
| 5:2,4: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑦 = 𝐵
↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵) )
| 6:5: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑦([𝐴 / 𝑥]𝑦
= 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵) )
| 7:6: | ⊢ ( 𝐴 ∈ 𝑉 ▶ {𝑦 ∣ [𝐴 / 𝑥]𝑦 =
𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} )
| 8:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ {𝑦 ∣ [𝐴 / 𝑥]𝑦 =
𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} )
| 9:7,8: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦
= 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} )
| 10:: | ⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
| 11:10: | ⊢ ∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
| 12:1,11: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝐵} = ⦋
𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} )
| 13:9,12: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝐵} = {
𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} )
| 14:: | ⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴
/ 𝑥⦌𝐵}
| 15:13,14: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝐵} = {
⦋𝐴 / 𝑥⦌𝐵} )
| qed:15: | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋
𝐴 / 𝑥⦌𝐵})
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵}) |
|
Theorem | csbxpgVD 41234 |
Virtual deduction proof of csbxp 5653.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbxp 5653 is csbxpgVD 41234 without virtual deductions and was
automatically derived from csbxpgVD 41234.
1:: | ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 )
| 2:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔
⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| 3:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌𝑤 = 𝑤 )
| 4:3: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 /
𝑥⦌𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| 5:2,4: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤
∈ ⦋𝐴 / 𝑥⦌𝐵) )
| 6:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔
⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶) )
| 7:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌𝑦 = 𝑦 )
| 8:7: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 /
𝑥⦌𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶) )
| 9:6,8: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦
∈ ⦋𝐴 / 𝑥⦌𝐶) )
| 10:5,9: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧
[𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧
𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)) )
| 11:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧
𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶)) )
| 12:10,11: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧
𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)) )
| 13:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑧 = 〈𝑤 ,
𝑦〉 ↔ 𝑧 = 〈𝑤, 𝑦〉) )
| 14:12,13: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (([𝐴 / 𝑥]𝑧 = 〈𝑤
, 𝑦〉 ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = 〈𝑤, 𝑦〉
∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| 15:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥](𝑧 = 〈𝑤
, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = 〈𝑤, 𝑦〉
∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) )
| 16:14,15: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥](𝑧 = 〈𝑤
, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = 〈𝑤, 𝑦〉 ∧
(𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| 17:16: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑦([𝐴 / 𝑥](𝑧 =
〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = 〈𝑤, 𝑦〉 ∧
(𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| 18:17: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∃𝑦[𝐴 / 𝑥](𝑧 =
〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = 〈𝑤, 𝑦〉 ∧
(𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| 19:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑦(𝑧 =
〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 =
〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) )
| 20:18,19: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑦(𝑧 =
〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = 〈𝑤, 𝑦〉 ∧
(𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| 21:20: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑤([𝐴 / 𝑥]∃𝑦(
𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 =
〈𝑤, 𝑦〉 ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| 22:21: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∃𝑤[𝐴 / 𝑥]∃𝑦(
𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 =
〈𝑤, 𝑦〉 ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| 23:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑤∃𝑦(
𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]∃𝑦
(𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) )
| 24:22,23: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑤∃𝑦(
𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 =
〈𝑤, 𝑦〉 ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| 25:24: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑧([𝐴 / 𝑥]∃𝑤∃
𝑦(𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 =
〈𝑤, 𝑦〉 ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) )
| 26:25: | ⊢ ( 𝐴 ∈ 𝑉 ▶ {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃
𝑦(𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(
𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
)
| 27:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃
𝑦(𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ [𝐴 / 𝑥]
∃𝑤∃𝑦(𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} )
| 28:26,27: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃
𝑦(𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(
𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
)
| 29:: | ⊢ {〈𝑤 , 𝑦〉 ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
= {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
| 30:: | ⊢ (𝐵 × 𝐶) = {〈𝑤 , 𝑦〉 ∣ (𝑤 ∈ 𝐵
∧ 𝑦 ∈ 𝐶)}
| 31:29,30: | ⊢ (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = 〈𝑤
, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
| 32:31: | ⊢ ∀𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 =
〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
| 33:1,32: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) =
⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ 𝐵 ∧
𝑦 ∈ 𝐶))} )
| 34:28,33: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) =
{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧
𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))} )
| 35:: | ⊢ {〈𝑤 , 𝑦〉 ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧
𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = 〈𝑤, 𝑦〉 ∧
(𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
| 36:: | ⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {
〈𝑤, 𝑦〉 ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)}
| 37:35,36: | ⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {𝑧
∣ ∃𝑤∃𝑦(𝑧 = 〈𝑤, 𝑦〉 ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧
𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
| 38:34,37: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) =
(⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) )
| qed:38: | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (
⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶)) |
|
Theorem | csbresgVD 41235 |
Virtual deduction proof of csbres 5859.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbres 5859 is csbresgVD 41235 without virtual deductions and was
automatically derived from csbresgVD 41235.
1:: | ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 )
| 2:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌V = V )
| 3:2: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 /
𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V) )
| 4:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐶 × V) =
(⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) )
| 5:3,4: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐶 × V) =
(⦋𝐴 / 𝑥⦌𝐶 × V) )
| 6:5: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 /
𝑥⦌(𝐶 × V)) =
(⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) )
| 7:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 ×
V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) )
| 8:6,7: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 ×
V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) )
| 9:: | ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
| 10:9: | ⊢ ∀𝑥(𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
| 11:1,10: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) =
⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) )
| 12:8,11: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶)
= (
⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) )
| 13:: | ⊢ (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = (
⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))
| 14:12,13: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) =
(
⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) )
| qed:14: | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (
⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶))
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)) |
|
Theorem | csbrngVD 41236 |
Virtual deduction proof of csbrn 6063.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbrn 6063 is csbrngVD 41236 without virtual deductions and was
automatically derived from csbrngVD 41236.
1:: | ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 )
| 2:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]〈𝑤 , 𝑦〉
∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| 3:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌〈𝑤 , 𝑦〉 =
〈𝑤, 𝑦〉 )
| 4:3: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌〈𝑤 , 𝑦〉
∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| 5:2,4: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]〈𝑤 , 𝑦〉
∈ 𝐵 ↔ 〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| 6:5: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑤([𝐴 / 𝑥]〈𝑤 ,
𝑦〉 ∈ 𝐵 ↔ 〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| 7:6: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∃𝑤[𝐴 / 𝑥]〈𝑤 ,
𝑦〉 ∈ 𝐵 ↔ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| 8:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∃𝑤[𝐴 / 𝑥]〈𝑤 ,
𝑦〉 ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵) )
| 9:7,8: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑤〈𝑤
, 𝑦〉 ∈ 𝐵 ↔ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| 10:9: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑦([𝐴 / 𝑥]∃𝑤
〈𝑤, 𝑦〉 ∈ 𝐵 ↔ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵) )
| 11:10: | ⊢ ( 𝐴 ∈ 𝑉 ▶ {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈
𝑤, 𝑦〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} )
| 12:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤
〈𝑤, 𝑦〉 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} )
| 13:11,12: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤
〈𝑤, 𝑦〉 ∈ 𝐵} = {𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} )
| 14:: | ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑤〈𝑤 , 𝑦〉 ∈ 𝐵}
| 15:14: | ⊢ ∀𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤〈𝑤 , 𝑦〉
∈ 𝐵}
| 16:1,15: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 /
𝑥⦌{𝑦 ∣ ∃𝑤〈𝑤, 𝑦〉 ∈ 𝐵} )
| 17:13,16: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌ran 𝐵 = {𝑦 ∣
∃𝑤〈𝑤, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵} )
| 18:: | ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤〈𝑤
, 𝑦〉 ∈ ⦋𝐴 / 𝑥⦌𝐵}
| 19:17,18: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋
𝐴 / 𝑥⦌𝐵 )
| qed:19: | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴
/ 𝑥⦌𝐵)
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵) |
|
Theorem | csbima12gALTVD 41237 |
Virtual deduction proof of csbima12 5950.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbima12 5950 is csbima12gALTVD 41237 without virtual deductions and was
automatically derived from csbima12gALTVD 41237.
1:: | ⊢ ( 𝐴 ∈ 𝐶 ▶ 𝐴 ∈ 𝐶 )
| 2:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) =
(
⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) )
| 3:2: | ⊢ ( 𝐴 ∈ 𝐶 ▶
ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵)
= ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) )
| 4:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶
⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵)
= ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) )
| 5:3,4: | ⊢ ( 𝐴 ∈ 𝐶 ▶
⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵)
= ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) )
| 6:: | ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
| 7:6: | ⊢ ∀𝑥(𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
| 8:1,7: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ⦋
𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) )
| 9:5,8: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) =
ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) )
| 10:: | ⊢ (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) = ran
(⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)
| 11:9,10: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (
⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) )
| qed:11: | ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋
𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)) |
|
Theorem | csbunigVD 41238 |
Virtual deduction proof of csbuni 4870.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbuni 4870 is csbunigVD 41238 without virtual deductions and was
automatically derived from csbunigVD 41238.
1:: | ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 )
| 2:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧
∈ 𝑦) )
| 3:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦
∈ ⦋𝐴 / 𝑥⦌𝐵) )
| 4:2,3: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧
[𝐴 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)) )
| 5:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧
𝑦 ∈ 𝐵) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵)) )
| 6:4,5: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧
𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)) )
| 7:6: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑦([𝐴 / 𝑥](𝑧 ∈
𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)) )
| 8:7: | ⊢ ( 𝐴 ∈ 𝑉 ▶ (∃𝑦[𝐴 / 𝑥](𝑧 ∈
𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)) )
| 9:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑦(𝑧 ∈
𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) )
| 10:8,9: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]∃𝑦(𝑧 ∈
𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)) )
| 11:10: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ∀𝑧([𝐴 / 𝑥]∃𝑦(
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)) )
| 12:11: | ⊢ ( 𝐴 ∈ 𝑉 ▶ {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(
𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧
𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)} )
| 13:1: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
)
| 14:12,13: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧
∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧
𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)} )
| 15:: | ⊢ ∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
| 16:15: | ⊢ ∀𝑥∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈
𝐵)}
| 17:1,16: | ⊢ ( 𝐴 ∈ 𝑉 ▶ [𝐴 / 𝑥]∪ 𝐵 = {𝑧 ∣
∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} )
| 18:1,17: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌∪ 𝐵 = ⦋𝐴 /
𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} )
| 19:14,18: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌∪ 𝐵 = {𝑧 ∣
∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)} )
| 20:: | ⊢ ∪ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦
∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}
| 21:19,20: | ⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴
/ 𝑥⦌𝐵 )
| qed:21: | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 /
𝑥⦌𝐵)
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪
𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵) |
|
Theorem | csbfv12gALTVD 41239 |
Virtual deduction proof of csbfv12 6716.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
csbfv12 6716 is csbfv12gALTVD 41239 without virtual deductions and was
automatically derived from csbfv12gALTVD 41239.
1:: | ⊢ ( 𝐴 ∈ 𝐶 ▶ 𝐴 ∈ 𝐶 )
| 2:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌{𝑦} = {
𝑦} )
| 3:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵
}) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) )
| 4:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌{𝐵} = {
⦋𝐴 / 𝑥⦌𝐵} )
| 5:4: | ⊢ ( 𝐴 ∈ 𝐶 ▶ (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴
/ 𝑥⦌{𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) )
| 6:3,5: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵
}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) )
| 7:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ([𝐴 / 𝑥](𝐹 “ {
𝐵}) = {𝑦} ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦}) )
| 8:6,2: | ⊢ ( 𝐴 ∈ 𝐶 ▶ (⦋𝐴 / 𝑥⦌(𝐹 “ {
𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})
= {𝑦}) )
| 9:7,8: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ([𝐴 / 𝑥](𝐹 “ {
𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})
)
| 10:9: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ∀𝑦([𝐴 / 𝑥](𝐹
“ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) =
{𝑦}) )
| 11:10: | ⊢ ( 𝐴 ∈ 𝐶 ▶ {𝑦 ∣ [𝐴 / 𝑥](𝐹
“ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) =
{𝑦}} )
| 12:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹
“ {𝐵}) = {𝑦}} = {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} )
| 13:11,12: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹
“ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) =
{𝑦
}} )
| 14:13: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (
𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “
{⦋𝐴 / 𝑥⦌𝐵}) =
{𝑦}} )
| 15:1: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (
𝐹 “ {𝐵}) = {𝑦}} = ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) =
{𝑦}} )
| 16:14,15: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (
𝐹 “ {𝐵}) = {𝑦}} =
∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) =
{𝑦}} )
| 17:: | ⊢ (𝐹‘𝐵) =
∪ {𝑦 ∣ (𝐹 “ {𝐵}) =
{𝑦}}
| 18:17: | ⊢ ∀𝑥(𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵
}) = {𝑦}}
| 19:1,18: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵)
= ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} )
| 20:16,19: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵)
= ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} )
| 21:: | ⊢ (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) =
∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}
| 22:20,21: | ⊢ ( 𝐴 ∈ 𝐶 ▶ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵)
= (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) )
| qed:22: | ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) =
(⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))
|
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)) |
|
Theorem | con5VD 41240 |
Virtual deduction proof of con5 40862.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
con5 40862 is con5VD 41240 without virtual deductions and was automatically
derived from con5VD 41240.
1:: | ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (𝜑 ↔ ¬ 𝜓) )
| 2:1: | ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜓 → 𝜑) )
| 3:2: | ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → ¬ ¬ 𝜓
) )
| 4:: | ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
| 5:3,4: | ⊢ ( (𝜑 ↔ ¬ 𝜓) ▶ (¬ 𝜑 → 𝜓) )
| qed:5: | ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓)) |
|
Theorem | relopabVD 41241 |
Virtual deduction proof of relopab 5699.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
relopab 5699 is relopabVD 41241 without virtual deductions and was
automatically derived from relopabVD 41241.
1:: | ⊢ ( 𝑦 = 𝑣 ▶ 𝑦 = 𝑣 )
| 2:1: | ⊢ ( 𝑦 = 𝑣 ▶ 〈𝑥 , 𝑦〉 = 〈𝑥 , 𝑣
〉 )
| 3:: | ⊢ ( 𝑦 = 𝑣 , 𝑥 = 𝑢 ▶ 𝑥 = 𝑢 )
| 4:3: | ⊢ ( 𝑦 = 𝑣 , 𝑥 = 𝑢 ▶ 〈𝑥 , 𝑣〉 = 〈
𝑢, 𝑣〉 )
| 5:2,4: | ⊢ ( 𝑦 = 𝑣 , 𝑥 = 𝑢 ▶ 〈𝑥 , 𝑦〉 = 〈
𝑢, 𝑣〉 )
| 6:5: | ⊢ ( 𝑦 = 𝑣 , 𝑥 = 𝑢 ▶ (𝑧 = 〈𝑥 , 𝑦
〉 → 𝑧 = 〈𝑢, 𝑣〉) )
| 7:6: | ⊢ ( 𝑦 = 𝑣 ▶ (𝑥 = 𝑢 → (𝑧 = 〈𝑥 ,
𝑦〉 → 𝑧 = 〈𝑢, 𝑣〉)) )
| 8:7: | ⊢ (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑧 = 〈𝑥 , 𝑦
〉 → 𝑧 = 〈𝑢, 𝑣〉)))
| 9:8: | ⊢ (∃𝑣𝑦 = 𝑣 → ∃𝑣(𝑥 = 𝑢 → (𝑧
= 〈𝑥, 𝑦〉 → 𝑧 = 〈𝑢, 𝑣〉)))
| 90:: | ⊢ (𝑣 = 𝑦 ↔ 𝑦 = 𝑣)
| 91:90: | ⊢ (∃𝑣𝑣 = 𝑦 ↔ ∃𝑣𝑦 = 𝑣)
| 92:: | ⊢ ∃𝑣𝑣 = 𝑦
| 10:91,92: | ⊢ ∃𝑣𝑦 = 𝑣
| 11:9,10: | ⊢ ∃𝑣(𝑥 = 𝑢 → (𝑧 = 〈𝑥 , 𝑦〉 →
𝑧 = 〈𝑢, 𝑣〉))
| 12:11: | ⊢ (𝑥 = 𝑢 → ∃𝑣(𝑧 = 〈𝑥 , 𝑦〉 →
𝑧 = 〈𝑢, 𝑣〉))
| 13:: | ⊢ (∃𝑣(𝑧 = 〈𝑥 , 𝑦〉 → 𝑧 = 〈𝑢
, 𝑣〉) → (𝑧 = 〈𝑥, 𝑦〉 → ∃𝑣𝑧 = 〈𝑢, 𝑣〉))
| 14:12,13: | ⊢ (𝑥 = 𝑢 → (𝑧 = 〈𝑥 , 𝑦〉 → ∃𝑣
𝑧 = 〈𝑢, 𝑣〉))
| 15:14: | ⊢ (∃𝑢𝑥 = 𝑢 → ∃𝑢(𝑧 = 〈𝑥 , 𝑦
〉 → ∃𝑣𝑧 = 〈𝑢, 𝑣〉))
| 150:: | ⊢ (𝑢 = 𝑥 ↔ 𝑥 = 𝑢)
| 151:150: | ⊢ (∃𝑢𝑢 = 𝑥 ↔ ∃𝑢𝑥 = 𝑢)
| 152:: | ⊢ ∃𝑢𝑢 = 𝑥
| 16:151,152: | ⊢ ∃𝑢𝑥 = 𝑢
| 17:15,16: | ⊢ ∃𝑢(𝑧 = 〈𝑥 , 𝑦〉 → ∃𝑣𝑧 = 〈
𝑢, 𝑣〉)
| 18:17: | ⊢ (𝑧 = 〈𝑥 , 𝑦〉 → ∃𝑢∃𝑣𝑧 = 〈
𝑢, 𝑣〉)
| 19:18: | ⊢ (∃𝑦𝑧 = 〈𝑥 , 𝑦〉 → ∃𝑦∃𝑢
∃𝑣𝑧 = 〈𝑢, 𝑣〉)
| 20:: | ⊢ (∃𝑦∃𝑢∃𝑣𝑧 = 〈𝑢 , 𝑣〉 →
∃𝑢∃𝑣𝑧 = 〈𝑢, 𝑣〉)
| 21:19,20: | ⊢ (∃𝑦𝑧 = 〈𝑥 , 𝑦〉 → ∃𝑢∃𝑣𝑧
= 〈𝑢, 𝑣〉)
| 22:21: | ⊢ (∃𝑥∃𝑦𝑧 = 〈𝑥 , 𝑦〉 → ∃𝑥
∃𝑢∃𝑣𝑧 = 〈𝑢, 𝑣〉)
| 23:: | ⊢ (∃𝑥∃𝑢∃𝑣𝑧 = 〈𝑢 , 𝑣〉 →
∃𝑢∃𝑣𝑧 = 〈𝑢, 𝑣〉)
| 24:22,23: | ⊢ (∃𝑥∃𝑦𝑧 = 〈𝑥 , 𝑦〉 → ∃𝑢
∃𝑣𝑧 = 〈𝑢, 𝑣〉)
| 25:24: | ⊢ {𝑧 ∣ ∃𝑥∃𝑦𝑧 = 〈𝑥 , 𝑦〉} ⊆
{𝑧 ∣ ∃𝑢∃𝑣𝑧 = 〈𝑢, 𝑣〉}
| 26:: | ⊢ 𝑥 ∈ V
| 27:: | ⊢ 𝑦 ∈ V
| 28:26,27: | ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
| 29:28: | ⊢ (𝑧 = 〈𝑥 , 𝑦〉 ↔ (𝑧 = 〈𝑥 , 𝑦
〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
| 30:29: | ⊢ (∃𝑦𝑧 = 〈𝑥 , 𝑦〉 ↔ ∃𝑦(𝑧 =
〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
| 31:30: | ⊢ (∃𝑥∃𝑦𝑧 = 〈𝑥 , 𝑦〉 ↔ ∃𝑥
∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
| 32:31: | ⊢ {𝑧 ∣ ∃𝑥∃𝑦𝑧 = 〈𝑥 , 𝑦〉} = {
𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
| 320:25,32: | ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥 , 𝑦〉 ∧
(𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = 〈𝑢, 𝑣〉}
| 33:: | ⊢ 𝑢 ∈ V
| 34:: | ⊢ 𝑣 ∈ V
| 35:33,34: | ⊢ (𝑢 ∈ V ∧ 𝑣 ∈ V)
| 36:35: | ⊢ (𝑧 = 〈𝑢 , 𝑣〉 ↔ (𝑧 = 〈𝑢 , 𝑣
〉 ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
| 37:36: | ⊢ (∃𝑣𝑧 = 〈𝑢 , 𝑣〉 ↔ ∃𝑣(𝑧 =
〈𝑢, 𝑣〉 ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
| 38:37: | ⊢ (∃𝑢∃𝑣𝑧 = 〈𝑢 , 𝑣〉 ↔ ∃𝑢
∃𝑣(𝑧 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
| 39:38: | ⊢ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = 〈𝑢 , 𝑣〉} = {
𝑧 ∣ ∃𝑢∃𝑣(𝑧 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
| 40:320,39: | ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥 , 𝑦〉 ∧
(𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = 〈𝑢, 𝑣〉 ∧
(𝑢 ∈ V ∧ 𝑣 ∈ V))}
| 41:: | ⊢ {〈𝑥 , 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V
)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))
}
| 42:: | ⊢ {〈𝑢 , 𝑣〉 ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V
)} = {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))
}
| 43:40,41,42: | ⊢ {〈𝑥 , 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V
)} ⊆ {〈𝑢, 𝑣〉 ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
| 44:: | ⊢ {〈𝑢 , 𝑣〉 ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V
)} = (V × V)
| 45:43,44: | ⊢ {〈𝑥 , 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V
)} ⊆ (V × V)
| 46:28: | ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
| 47:46: | ⊢ {〈𝑥 , 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥 , 𝑦〉
∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
| 48:45,47: | ⊢ {〈𝑥 , 𝑦〉 ∣ 𝜑} ⊆ (V × V)
| qed:48: | ⊢ Rel {〈𝑥 , 𝑦〉 ∣ 𝜑}
|
(Contributed by Alan Sare, 9-Jul-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ Rel
{〈𝑥, 𝑦〉 ∣ 𝜑} |
|
Theorem | 19.41rgVD 41242 |
Virtual deduction proof of 19.41rg 40890.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. 19.41rg 40890
is 19.41rgVD 41242 without virtual deductions and was automatically derived
from 19.41rgVD 41242. (Contributed by Alan Sare, 8-Feb-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
1:: | ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
| 2:1: | ⊢ ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (
𝜑 ∧ 𝜓))))
| 3:2: | ⊢ ∀𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑
→ (𝜑 ∧ 𝜓))))
| 4:3: | ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 →
∀𝑥(𝜑 → (𝜑 ∧ 𝜓))))
| 5:: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ ∀𝑥(𝜓
→ ∀𝑥𝜓) )
| 6:4,5: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ (∀𝑥𝜓
→ ∀𝑥(𝜑 → (𝜑 ∧ 𝜓))) )
| 7:: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) , ∀𝑥𝜓 ▶
∀𝑥𝜓 )
| 8:6,7: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) , ∀𝑥𝜓 ▶
∀𝑥(𝜑 → (𝜑 ∧ 𝜓)) )
| 9:8: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) , ∀𝑥𝜓 ▶
(∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)) )
| 10:9: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ (∀𝑥𝜓
→ (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) )
| 11:5: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ (𝜓 → ∀
𝑥𝜓) )
| 12:10,11: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ (𝜓 → (
∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) )
| 13:12: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ (∃𝑥𝜑
→ (𝜓 → ∃𝑥(𝜑 ∧ 𝜓))) )
| 14:13: | ⊢ ( ∀𝑥(𝜓 → ∀𝑥𝜓) ▶ ((∃𝑥
𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) )
| qed:14: | ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑
∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
|
|
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))) |
|
Theorem | 2pm13.193VD 41243 |
Virtual deduction proof of 2pm13.193 40892.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
2pm13.193 40892 is 2pm13.193VD 41243 without virtual deductions and was
automatically derived from 2pm13.193VD 41243. (Contributed by Alan Sare,
8-Feb-2014.) (Proof modification is discouraged.)
(New usage is discouraged.)
1:: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) )
| 2:1: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) )
| 3:2: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ 𝑥 = 𝑢 )
| 4:1: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑 )
| 5:3,4: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢) )
| 6:5: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ ([𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢) )
| 7:6: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ [𝑣 / 𝑦]𝜑 )
| 8:2: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ 𝑦 = 𝑣 )
| 9:7,8: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ ([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣) )
| 10:9: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ (𝜑 ∧ 𝑦 = 𝑣) )
| 11:10: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ 𝜑 )
| 12:2,11: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) )
| 13:12: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣
/ 𝑦]𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| 14:: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ ((
𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) )
| 15:14: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ (𝑥
= 𝑢 ∧ 𝑦 = 𝑣) )
| 16:15: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ 𝑦 =
𝑣 )
| 17:14: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ 𝜑
)
| 18:16,17: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ (
𝜑 ∧ 𝑦 = 𝑣) )
| 19:18: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ ([
𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣) )
| 20:15: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ 𝑥 =
𝑢 )
| 21:19: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ [𝑣
/ 𝑦]𝜑 )
| 22:20,21: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ ([
𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢) )
| 23:22: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ ([
𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢) )
| 24:23: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ [𝑢
/ 𝑥][𝑣 / 𝑦]𝜑 )
| 25:15,24: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ▶ ((
𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) )
| 26:25: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → ((𝑥
= 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| qed:13,26: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣
/ 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
|
|
⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) |
|
Theorem | hbimpgVD 41244 |
Virtual deduction proof of hbimpg 40894.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. hbimpg 40894
is hbimpgVD 41244 without virtual deductions and was automatically derived
from hbimpgVD 41244. (Contributed by Alan Sare, 8-Feb-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
1:: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 →
∀𝑥𝜓)) )
| 2:1: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ ∀𝑥(𝜑 → ∀𝑥𝜑) )
| 3:: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)), ¬ 𝜑 ▶ ¬ 𝜑 )
| 4:2: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ ∀𝑥(¬ 𝜑 → ∀𝑥¬ 𝜑) )
| 5:4: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ (¬ 𝜑 → ∀𝑥¬ 𝜑) )
| 6:3,5: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)), ¬ 𝜑 ▶ ∀𝑥¬ 𝜑 )
| 7:: | ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
| 8:7: | ⊢ (∀𝑥¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))
| 9:6,8: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)), ¬ 𝜑 ▶ ∀𝑥(𝜑 → 𝜓) )
| 10:9: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ (¬ 𝜑 → ∀𝑥(𝜑 → 𝜓)) )
| 11:: | ⊢ (𝜓 → (𝜑 → 𝜓))
| 12:11: | ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → 𝜓))
| 13:1: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ ∀𝑥(𝜓 → ∀𝑥𝜓) )
| 14:13: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ (𝜓 → ∀𝑥𝜓) )
| 15:14,12: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ (𝜓 → ∀𝑥(𝜑 → 𝜓)) )
| 16:10,15: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ ((¬ 𝜑 ∨ 𝜓) → ∀𝑥(𝜑 → 𝜓)) )
| 17:: | ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
| 18:16,17: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) )
| 19:: | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑥(
𝜑 → ∀𝑥𝜑))
| 20:: | ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ∀𝑥∀𝑥(
𝜓 → ∀𝑥𝜓))
| 21:19,20: | ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) → ∀𝑥(∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 →
∀𝑥𝜓)))
| 22:21,18: | ⊢ ( (∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) ▶ ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) )
| qed:22: | ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓
→ ∀𝑥𝜓)) → ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
|
|
⊢
((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))) |
|
Theorem | hbalgVD 41245 |
Virtual deduction proof of hbalg 40895.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. hbalg 40895
is hbalgVD 41245 without virtual deductions and was automatically derived
from hbalgVD 41245. (Contributed by Alan Sare, 8-Feb-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
1:: | ⊢ ( ∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦(𝜑
→ ∀𝑥𝜑) )
| 2:1: | ⊢ ( ∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (∀𝑦𝜑
→ ∀𝑦∀𝑥𝜑) )
| 3:: | ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
| 4:2,3: | ⊢ ( ∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (∀𝑦𝜑
→ ∀𝑥∀𝑦𝜑) )
| 5:: | ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑦(
𝜑 → ∀𝑥𝜑))
| 6:5,4: | ⊢ ( ∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦(∀
𝑦𝜑 → ∀𝑥∀𝑦𝜑) )
| qed:6: | ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦
𝜑 → ∀𝑥∀𝑦𝜑))
|
|
⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) |
|
Theorem | hbexgVD 41246 |
Virtual deduction proof of hbexg 40896.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. hbexg 40896
is hbexgVD 41246 without virtual deductions and was automatically derived
from hbexgVD 41246. (Contributed by Alan Sare, 8-Feb-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
1:: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
∀𝑦(𝜑 → ∀𝑥𝜑) )
| 2:1: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦
∀𝑥(𝜑 → ∀𝑥𝜑) )
| 3:2: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
(𝜑 → ∀𝑥𝜑) )
| 4:3: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
(¬ 𝜑 → ∀𝑥¬ 𝜑) )
| 5:: | ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦
∀𝑥(𝜑 → ∀𝑥𝜑))
| 6:: | ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦
∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
| 7:5: | ⊢ (∀𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ↔
∀𝑦∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
| 8:5,6,7: | ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦
∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
| 9:8,4: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦
∀𝑥(¬ 𝜑 → ∀𝑥¬ 𝜑) )
| 10:9: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
∀𝑦(¬ 𝜑 → ∀𝑥¬ 𝜑) )
| 11:10: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦
(¬ 𝜑 → ∀𝑥¬ 𝜑) )
| 12:11: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦
(∀𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑) )
| 13:12: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (∀
𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑) )
| 14:: | ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥
∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
| 15:13,14: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
(∀𝑦¬ 𝜑 → ∀𝑥∀𝑦¬ 𝜑) )
| 16:15: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
(¬ ∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑) )
| 17:16: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (¬
∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑) )
| 18:: | ⊢ (∃𝑦𝜑 ↔ ¬ ∀𝑦¬ 𝜑)
| 19:17,18: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (∃
𝑦𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑) )
| 20:18: | ⊢ (∀𝑥∃𝑦𝜑 ↔ ∀𝑥¬ ∀𝑦¬ 𝜑)
| 21:19,20: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (∃
𝑦𝜑 → ∀𝑥∃𝑦𝜑) )
| 22:8,21: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦
(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) )
| 23:14,22: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) )
| qed:23: | ⊢ ( ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑥
∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) )
|
|
⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) |
|
Theorem | ax6e2eqVD 41247* |
The following User's Proof is a Virtual Deduction proof (see wvd1 40909)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2eq 40897 is ax6e2eqVD 41247 without virtual
deductions and was automatically derived from ax6e2eqVD 41247.
(Contributed by Alan Sare, 25-Mar-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
1:: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∀𝑥𝑥 = 𝑦 )
| 2:: | ⊢ ( ∀𝑥𝑥 = 𝑦 , 𝑥 = 𝑢 ▶ 𝑥 = 𝑢 )
| 3:1: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ 𝑥 = 𝑦 )
| 4:2,3: | ⊢ ( ∀𝑥𝑥 = 𝑦 , 𝑥 = 𝑢 ▶ 𝑦 = 𝑢 )
| 5:2,4: | ⊢ ( ∀𝑥𝑥 = 𝑦 , 𝑥 = 𝑢 ▶ (𝑥 = 𝑢 ∧ 𝑦
= 𝑢) )
| 6:5: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ (𝑥 = 𝑢 → (𝑥 = 𝑢 ∧
𝑦 = 𝑢)) )
| 7:6: | ⊢ (∀𝑥𝑥 = 𝑦 → (𝑥 = 𝑢 → (𝑥 = 𝑢 ∧ 𝑦
= 𝑢)))
| 8:7: | ⊢ (∀𝑥∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (
𝑥 = 𝑢 ∧ 𝑦 = 𝑢)))
| 9:: | ⊢ (∀𝑥𝑥 = 𝑦 ↔ ∀𝑥∀𝑥𝑥 = 𝑦)
| 10:8,9: | ⊢ (∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → (𝑥 = 𝑢
∧ 𝑦 = 𝑢)))
| 11:1,10: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∀𝑥(𝑥 = 𝑢 → (𝑥 =
𝑢 ∧ 𝑦 = 𝑢)) )
| 12:11: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ (∃𝑥𝑥 = 𝑢 → ∃𝑥
(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)) )
| 13:: | ⊢ ∃𝑥𝑥 = 𝑢
| 14:13,12: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢
) )
| 140:14: | ⊢ (∀𝑥𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)
)
| 141:140: | ⊢ (∀𝑥𝑥 = 𝑦 → ∀𝑥∃𝑥(𝑥 = 𝑢 ∧ 𝑦
= 𝑢))
| 15:1,141: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∀𝑥∃𝑥(𝑥 = 𝑢 ∧
𝑦 = 𝑢) )
| 16:1,15: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∀𝑦∃𝑥(𝑥 = 𝑢 ∧
𝑦 = 𝑢) )
| 17:16: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∃𝑦∃𝑥(𝑥 = 𝑢 ∧
𝑦 = 𝑢) )
| 18:17: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧
𝑦 = 𝑢) )
| 19:: | ⊢ ( 𝑢 = 𝑣 ▶ 𝑢 = 𝑣 )
| 20:: | ⊢ ( 𝑢 = 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑢) ▶ (𝑥 =
𝑢 ∧ 𝑦 = 𝑢) )
| 21:20: | ⊢ ( 𝑢 = 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑢) ▶ 𝑦 = 𝑢
)
| 22:19,21: | ⊢ ( 𝑢 = 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑢) ▶ 𝑦 = 𝑣
)
| 23:20: | ⊢ ( 𝑢 = 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑢) ▶ 𝑥 = 𝑢
)
| 24:22,23: | ⊢ ( 𝑢 = 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑢) ▶ (𝑥 =
𝑢 ∧ 𝑦 = 𝑣) )
| 25:24: | ⊢ ( 𝑢 = 𝑣 ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → (
𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| 26:25: | ⊢ ( 𝑢 = 𝑣 ▶ ∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑢)
→ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| 27:26: | ⊢ ( 𝑢 = 𝑣 ▶ (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢)
→ ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| 28:27: | ⊢ ( 𝑢 = 𝑣 ▶ ∀𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 =
𝑢) → ∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| 29:28: | ⊢ ( 𝑢 = 𝑣 ▶ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 =
𝑢) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| 30:29: | ⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑢
) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
| 31:18,30: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ (𝑢 = 𝑣 → ∃𝑥∃𝑦
(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| qed:31: | ⊢ (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(
𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
|
|
⊢ (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) |
|
Theorem | ax6e2ndVD 41248* |
The following User's Proof is a Virtual Deduction proof (see wvd1 40909)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2nd 40898 is ax6e2ndVD 41248 without virtual
deductions and was automatically derived from ax6e2ndVD 41248.
(Contributed by Alan Sare, 25-Mar-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
1:: | ⊢ ∃𝑦𝑦 = 𝑣
| 2:: | ⊢ 𝑢 ∈ V
| 3:1,2: | ⊢ (𝑢 ∈ V ∧ ∃𝑦𝑦 = 𝑣)
| 4:3: | ⊢ ∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣)
| 5:: | ⊢ (𝑢 ∈ V ↔ ∃𝑥𝑥 = 𝑢)
| 6:5: | ⊢ ((𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (∃𝑥𝑥 =
𝑢 ∧ 𝑦 = 𝑣))
| 7:6: | ⊢ (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦
(∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
| 8:4,7: | ⊢ ∃𝑦(∃𝑥𝑥 = 𝑢 ∧ 𝑦 = 𝑣)
| 9:: | ⊢ (𝑧 = 𝑣 → ∀𝑥𝑧 = 𝑣)
| 10:: | ⊢ (𝑦 = 𝑣 → ∀𝑧𝑦 = 𝑣)
| 11:: | ⊢ ( 𝑧 = 𝑦 ▶ 𝑧 = 𝑦 )
| 12:11: | ⊢ ( 𝑧 = 𝑦 ▶ (𝑧 = 𝑣 ↔ 𝑦 = 𝑣) )
| 120:11: | ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣))
| 13:9,10,120: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥𝑦
= 𝑣))
| 14:: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ¬ ∀𝑥𝑥 = 𝑦 )
| 15:14,13: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ (𝑦 = 𝑣 → ∀𝑥
𝑦 = 𝑣) )
| 16:15: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥𝑦
= 𝑣))
| 17:16: | ⊢ (∀𝑥¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣
→ ∀𝑥𝑦 = 𝑣))
| 18:: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥¬ ∀𝑥𝑥 = 𝑦
)
| 19:17,18: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀
𝑥𝑦 = 𝑣))
| 20:14,19: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ∀𝑥(𝑦 = 𝑣 →
∀𝑥𝑦 = 𝑣) )
| 21:20: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ((∃𝑥𝑥 = 𝑢
∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| 22:21: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ((∃𝑥𝑥 = 𝑢 ∧
𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
| 23:22: | ⊢ (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦((∃𝑥
𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
| 24:: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥𝑥 = 𝑦
)
| 25:23,24: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦((∃𝑥𝑥 =
𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
| 26:14,25: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ∀𝑦((∃𝑥𝑥
= 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| 27:26: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ (∃𝑦(∃𝑥𝑥
= 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) )
| 28:8,27: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ∃𝑦∃𝑥(𝑥 =
𝑢 ∧ 𝑦 = 𝑣) )
| 29:28: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ∃𝑥∃𝑦(𝑥 =
𝑢 ∧ 𝑦 = 𝑣) )
| qed:29: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢
∧ 𝑦 = 𝑣))
|
|
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
|
Theorem | ax6e2ndeqVD 41249* |
The following User's Proof is a Virtual Deduction proof (see wvd1 40909)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. ax6e2eq 40897 is ax6e2ndeqVD 41249 without virtual
deductions and was automatically derived from ax6e2ndeqVD 41249.
(Contributed by Alan Sare, 25-Mar-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
1:: | ⊢ ( 𝑢 ≠ 𝑣 ▶ 𝑢 ≠ 𝑣 )
| 2:: | ⊢ ( 𝑢 ≠ 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ▶ (
𝑥 = 𝑢 ∧ 𝑦 = 𝑣) )
| 3:2: | ⊢ ( 𝑢 ≠ 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ▶ 𝑥
= 𝑢 )
| 4:1,3: | ⊢ ( 𝑢 ≠ 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ▶ 𝑥
≠ 𝑣 )
| 5:2: | ⊢ ( 𝑢 ≠ 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ▶ 𝑦
= 𝑣 )
| 6:4,5: | ⊢ ( 𝑢 ≠ 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ▶ 𝑥
≠ 𝑦 )
| 7:: | ⊢ (∀𝑥𝑥 = 𝑦 → 𝑥 = 𝑦)
| 8:7: | ⊢ (¬ 𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦)
| 9:: | ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦)
| 10:8,9: | ⊢ (𝑥 ≠ 𝑦 → ¬ ∀𝑥𝑥 = 𝑦)
| 11:6,10: | ⊢ ( 𝑢 ≠ 𝑣 , (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ▶
¬ ∀𝑥𝑥 = 𝑦 )
| 12:11: | ⊢ ( 𝑢 ≠ 𝑣 ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣)
→ ¬ ∀𝑥𝑥 = 𝑦) )
| 13:12: | ⊢ ( 𝑢 ≠ 𝑣 ▶ ∀𝑥((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) → ¬ ∀𝑥𝑥 = 𝑦) )
| 14:13: | ⊢ ( 𝑢 ≠ 𝑣 ▶ (∃𝑥(𝑥 = 𝑢 ∧ 𝑦 =
𝑣) → ∃𝑥¬ ∀𝑥𝑥 = 𝑦) )
| 15:: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑥¬ ∀𝑥𝑥 = 𝑦
)
| 19:15: | ⊢ (∃𝑥¬ ∀𝑥𝑥 = 𝑦 ↔ ¬ ∀𝑥𝑥 =
𝑦)
| 20:14,19: | ⊢ ( 𝑢 ≠ 𝑣 ▶ (∃𝑥(𝑥 = 𝑢 ∧ 𝑦 =
𝑣) → ¬ ∀𝑥𝑥 = 𝑦) )
| 21:20: | ⊢ ( 𝑢 ≠ 𝑣 ▶ ∀𝑦(∃𝑥(𝑥 = 𝑢 ∧
𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦) )
| 22:21: | ⊢ ( 𝑢 ≠ 𝑣 ▶ (∃𝑦∃𝑥(𝑥 = 𝑢 ∧
𝑦 = 𝑣) → ∃𝑦¬ ∀𝑥𝑥 = 𝑦) )
| 23:: | ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ↔ ∃
𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
| 24:22,23: | ⊢ ( 𝑢 ≠ 𝑣 ▶ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧
𝑦 = 𝑣) → ∃𝑦¬ ∀𝑥𝑥 = 𝑦) )
| 25:: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥𝑥 = 𝑦
)
| 26:25: | ⊢ (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ∃𝑦∀𝑦¬
∀𝑥𝑥 = 𝑦)
| 260:: | ⊢ (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦∀𝑦¬
∀𝑥𝑥 = 𝑦)
| 27:260: | ⊢ (∃𝑦∀𝑦¬ ∀𝑥𝑥 = 𝑦 ↔ ∀𝑦¬
∀𝑥𝑥 = 𝑦)
| 270:26,27: | ⊢ (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ∀𝑦¬ ∀𝑥
𝑥 = 𝑦)
| 28:: | ⊢ (∀𝑦¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦
)
| 29:270,28: | ⊢ (∃𝑦¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑥𝑥 = 𝑦
)
| 30:24,29: | ⊢ ( 𝑢 ≠ 𝑣 ▶ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧
𝑦 = 𝑣) → ¬ ∀𝑥𝑥 = 𝑦) )
| 31:30: | ⊢ ( 𝑢 ≠ 𝑣 ▶ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧
𝑦 = 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)) )
| 32:31: | ⊢ (𝑢 ≠ 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦
= 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)))
| 33:: | ⊢ ( 𝑢 = 𝑣 ▶ 𝑢 = 𝑣 )
| 34:33: | ⊢ ( 𝑢 = 𝑣 ▶ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦
= 𝑣) → 𝑢 = 𝑣) )
| 35:34: | ⊢ ( 𝑢 = 𝑣 ▶ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦
= 𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)) )
| 36:35: | ⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 =
𝑣) → (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)))
| 37:: | ⊢ (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣)
| 38:32,36,37: | ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (
¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣))
| 39:: | ⊢ (∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦
(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
| 40:: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢
∧ 𝑦 = 𝑣))
| 41:40: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃
𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
| 42:: | ⊢ (∀𝑥𝑥 = 𝑦 ∨ ¬ ∀𝑥𝑥 = 𝑦)
| 43:39,41,42: | ⊢ (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣
))
| 44:40,43: | ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ∃𝑥
∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
| qed:38,44: | ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥
∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
|
|
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
|
Theorem | 2sb5ndVD 41250* |
The following User's Proof is a Virtual Deduction proof (see wvd1 40909)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. 2sb5nd 40900 is 2sb5ndVD 41250 without virtual
deductions and was automatically derived from 2sb5ndVD 41250.
(Contributed by Alan Sare, 30-Apr-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
1:: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][
𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| 2:1: | ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 /
𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| 3:: | ⊢ ([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣 / 𝑦]𝜑)
| 4:3: | ⊢ [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 → ∀𝑦[𝑣
/ 𝑦]𝜑)
| 5:4: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥]
∀𝑦[𝑣 / 𝑦]𝜑)
| 6:: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ¬ ∀𝑥𝑥 = 𝑦 )
| 7:: | ⊢ (∀𝑦𝑦 = 𝑥 → ∀𝑥𝑥 = 𝑦)
| 8:7: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ¬ ∀𝑦𝑦 = 𝑥)
| 9:6,8: | ⊢ ( ¬ ∀𝑥𝑥 = 𝑦 ▶ ¬ ∀𝑦𝑦 = 𝑥 )
| 10:9: | ⊢ ([𝑢 / 𝑥]∀𝑦[𝑣 / 𝑦]𝜑 ↔ ∀
𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
| 11:5,10: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢
/ 𝑥][𝑣 / 𝑦]𝜑)
| 12:11: | ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 /
𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| 13:: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑥[𝑢
/ 𝑥][𝑣 / 𝑦]𝜑)
| 14:: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ∀𝑥𝑥 = 𝑦 )
| 15:14: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ (∀𝑥[𝑢 / 𝑥][
𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) )
| 16:13,15: | ⊢ ( ∀𝑥𝑥 = 𝑦 ▶ ([𝑢 / 𝑥][𝑣 / 𝑦
]𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) )
| 17:16: | ⊢ (∀𝑥𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]
𝜑 → ∀𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| 19:12,17: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ∀𝑦[𝑢
/ 𝑥][𝑣 / 𝑦]𝜑)
| 20:19: | ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 /
𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| 21:2,20: | ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)
↔ (∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| 22:21: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
𝜑) ↔ ∃𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| 23:13: | ⊢ (∃𝑥(∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [
𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
[𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| 24:22,23: | ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [
𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| 240:24: | ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔
(∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
𝜑)))
| 241:: | ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)) ↔
(∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
| 242:241,240: | ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [
𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))
| 243:: | ⊢ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (
[𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
𝜑))) ↔ ((∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
[𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))))
| 25:242,243: | ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ([
𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))
| 26:: | ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥
∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
| qed:25,26: | ⊢ ((¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢
/ 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))
|
|
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) |
|
Theorem | 2uasbanhVD 41251* |
The following User's Proof is a Virtual Deduction proof (see wvd1 40909)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. 2uasbanh 40901 is 2uasbanhVD 41251 without
virtual deductions and was automatically derived from 2uasbanhVD 41251.
(Contributed by Alan Sare, 31-May-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
h1:: | ⊢ (𝜒 ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
| 100:1: | ⊢ (𝜒 → (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
| 2:100: | ⊢ ( 𝜒 ▶ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦
= 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) )
| 3:2: | ⊢ ( 𝜒 ▶ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ 𝜑) )
| 4:3: | ⊢ ( 𝜒 ▶ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣
) )
| 5:4: | ⊢ ( 𝜒 ▶ (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)
)
| 6:5: | ⊢ ( 𝜒 ▶ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑
↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) )
| 7:3,6: | ⊢ ( 𝜒 ▶ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑 )
| 8:2: | ⊢ ( 𝜒 ▶ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ 𝜓) )
| 9:5: | ⊢ ( 𝜒 ▶ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜓
↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) )
| 10:8,9: | ⊢ ( 𝜒 ▶ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓 )
| 101:: | ⊢ ([𝑣 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑣 /
𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
| 102:101: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓)
↔ [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
| 103:: | ⊢ ([𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦
]𝜓) ↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓))
| 104:102,103: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓)
↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓))
| 11:7,10,104: | ⊢ ( 𝜒 ▶ [𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧
𝜓) )
| 110:5: | ⊢ ( 𝜒 ▶ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑
∧ 𝜓) ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓))) )
| 12:11,110: | ⊢ ( 𝜒 ▶ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ (𝜑 ∧ 𝜓)) )
| 120:12: | ⊢ (𝜒 → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣
) ∧ (𝜑 ∧ 𝜓)))
| 13:1,120: | ⊢ ((∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) →
∃𝑥∃𝑦((𝑥 = 𝑢
∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)))
| 14:: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) )
| 15:14: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) )
| 16:14: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ (𝜑 ∧ 𝜓) )
| 17:16: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ 𝜑 )
| 18:15,17: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) )
| 19:18: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| 20:19: | ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑
∧ 𝜓)) → ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| 21:20: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
𝜑 ∧ 𝜓)) → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| 22:16: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ 𝜓 )
| 23:15,22: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓) )
| 24:23: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))
| 25:24: | ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑
∧ 𝜓)) → ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))
| 26:25: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
𝜑 ∧ 𝜓)) → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))
| 27:21,26: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
𝜑 ∧ 𝜓)) → (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧
∃𝑥∃𝑦(
(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
| qed:13,27: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧
∃𝑥∃𝑦(
(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
|
|
⊢ (𝜒 ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) ⇒ ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) |
|
Theorem | e2ebindVD 41252 |
The following User's Proof is a Virtual Deduction proof (see wvd1 40909)
completed automatically by a Metamath tools program invoking mmj2 and the
Metamath Proof Assistant. e2ebind 40903 is e2ebindVD 41252 without virtual
deductions and was automatically derived from e2ebindVD 41252.
1:: | ⊢ (𝜑 ↔ 𝜑)
| 2:1: | ⊢ (∀𝑦𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
| 3:2: | ⊢ (∀𝑦𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑
))
| 4:: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ ∀𝑦𝑦 = 𝑥 )
| 5:3,4: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑦𝜑 ↔ ∃𝑥
𝜑) )
| 6:: | ⊢ (∀𝑦𝑦 = 𝑥 → ∀𝑦∀𝑦𝑦 = 𝑥)
| 7:5,6: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ ∀𝑦(∃𝑦𝜑 ↔
∃𝑥𝜑) )
| 8:7: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑦∃𝑦𝜑 ↔
∃𝑦∃𝑥𝜑) )
| 9:: | ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
| 10:8,9: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑦∃𝑦𝜑 ↔
∃𝑥∃𝑦𝜑) )
| 11:: | ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
| 12:11: | ⊢ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑)
| 13:10,12: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑥∃𝑦𝜑 ↔
∃𝑦𝜑) )
| 14:13: | ⊢ (∀𝑦𝑦 = 𝑥 → (∃𝑥∃𝑦𝜑 ↔ ∃
𝑦𝜑))
| 15:: | ⊢ (∀𝑦𝑦 = 𝑥 ↔ ∀𝑥𝑥 = 𝑦)
| qed:14,15: | ⊢ (∀𝑥𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃
𝑦𝜑))
|
(Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
|
20.36.8 Virtual Deduction transcriptions of
textbook proofs
|
|
Theorem | sb5ALTVD 41253* |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Unit 20
Excercise 3.a., which is sb5 2275, found in the "Answers to Starred
Exercises" on page 457 of "Understanding Symbolic Logic", Fifth
Edition (2008), by Virginia Klenk. The same proof may also be
interpreted as a Virtual Deduction Hilbert-style axiomatic proof. It
was completed automatically by the tools program completeusersproof.cmd,
which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof
Assistant. sb5ALT 40865 is sb5ALTVD 41253 without virtual deductions and
was automatically derived from sb5ALTVD 41253.
1:: | ⊢ ( [𝑦 / 𝑥]𝜑 ▶ [𝑦 / 𝑥]𝜑 )
| 2:: | ⊢ [𝑦 / 𝑥]𝑥 = 𝑦
| 3:1,2: | ⊢ ( [𝑦 / 𝑥]𝜑 ▶ [𝑦 / 𝑥](𝑥 = 𝑦
∧ 𝜑) )
| 4:3: | ⊢ ( [𝑦 / 𝑥]𝜑 ▶ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑
) )
| 5:4: | ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
)
| 6:: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ▶ ∃𝑥(𝑥 =
𝑦 ∧ 𝜑) )
| 7:: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑
) ▶ (𝑥 = 𝑦 ∧ 𝜑) )
| 8:7: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑
) ▶ 𝜑 )
| 9:7: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑
) ▶ 𝑥 = 𝑦 )
| 10:8,9: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑
) ▶ [𝑦 / 𝑥]𝜑 )
| 101:: | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
| 11:101,10: | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑
)
| 12:5,11: | ⊢ (([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑
)) ∧ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑))
| qed:12: | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
)
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
|
Theorem | vk15.4jVD 41254 |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Unit 15
Excercise 4.f. found in the "Answers to Starred Exercises" on page 442
of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia
Klenk. The same proof may also be interpreted to be a Virtual Deduction
Hilbert-style axiomatic proof. It was completed automatically by the
tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2
and Norm Megill's Metamath Proof Assistant. vk15.4j 40868 is vk15.4jVD 41254
without virtual deductions and was automatically derived
from vk15.4jVD 41254. Step numbers greater than 25 are additional steps
necessary for the sequent calculus proof not contained in the
Fitch-style proof. Otherwise, step i of the User's Proof corresponds to
step i of the Fitch-style proof.
h1:: | ⊢ ¬ (∃𝑥¬ 𝜑 ∧ ∃𝑥(𝜓 ∧
¬ 𝜒))
| h2:: | ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏
))
| h3:: | ⊢ ¬ ∀𝑥(𝜏 → 𝜑)
| 4:: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ∃𝑥¬
𝜃 )
| 5:4: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∀𝑥𝜃 )
| 6:3: | ⊢ ∃𝑥(𝜏 ∧ ¬ 𝜑)
| 7:: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ (𝜏 ∧ ¬ 𝜑) )
| 8:7: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ 𝜏 )
| 9:7: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ¬ 𝜑 )
| 10:5: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ 𝜃 )
| 11:10,8: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ (𝜃 ∧ 𝜏) )
| 12:11: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ∃𝑥(𝜃 ∧ 𝜏) )
| 13:12: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏) )
| 14:2,13: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ¬ ∀𝑥𝜒 )
| 140:: | ⊢ (∃𝑥¬ 𝜃 → ∀𝑥∃𝑥¬ 𝜃
)
| 141:140: | ⊢ (¬ ∃𝑥¬ 𝜃 → ∀𝑥¬ ∃𝑥
¬ 𝜃)
| 142:: | ⊢ (∀𝑥𝜒 → ∀𝑥∀𝑥𝜒)
| 143:142: | ⊢ (¬ ∀𝑥𝜒 → ∀𝑥¬ ∀𝑥𝜒
)
| 144:6,14,141,143: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ∀𝑥𝜒
)
| 15:1: | ⊢ (¬ ∃𝑥¬ 𝜑 ∨ ¬ ∃𝑥(𝜓
∧ ¬ 𝜒))
| 16:9: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ∃𝑥¬ 𝜑 )
| 161:: | ⊢ (∃𝑥¬ 𝜑 → ∀𝑥∃𝑥¬ 𝜑
)
| 162:6,16,141,161: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∃𝑥¬ 𝜑
)
| 17:162: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ¬ ∃𝑥
¬ 𝜑 )
| 18:15,17: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ∃𝑥(
𝜓 ∧ ¬ 𝜒) )
| 19:18: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∀𝑥(𝜓
→ 𝜒) )
| 20:144: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∃𝑥¬ 𝜒
)
| 21:: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , ¬ 𝜒 ▶ ¬
𝜒 )
| 22:19: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ (𝜓 → 𝜒
) )
| 23:21,22: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , ¬ 𝜒 ▶ ¬
𝜓 )
| 24:23: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , ¬ 𝜒 ▶ ∃
𝑥¬ 𝜓 )
| 240:: | ⊢ (∃𝑥¬ 𝜓 → ∀𝑥∃𝑥¬ 𝜓
)
| 241:20,24,141,240: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∃𝑥¬ 𝜓
)
| 25:241: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ∀𝑥𝜓
)
| qed:25: | ⊢ (¬ ∃𝑥¬ 𝜃 → ¬ ∀𝑥𝜓)
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ¬
(∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) & ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏)) & ⊢ ¬
∀𝑥(𝜏 → 𝜑) ⇒ ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓) |
|
Theorem | notnotrALTVD 41255 |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Theorem 5 of
Section 14 of [Margaris] p. 59 (which is notnotr 132). The same proof
may also be interpreted as a Virtual Deduction Hilbert-style
axiomatic proof. It was completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. notnotrALT 40869 is notnotrALTVD 41255
without virtual deductions and was automatically derived
from notnotrALTVD 41255. Step i of the User's Proof corresponds to
step i of the Fitch-style proof.
1:: | ⊢ ( ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜑 )
| 2:: | ⊢ (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
| 3:1: | ⊢ ( ¬ ¬ 𝜑 ▶ (¬ 𝜑 → ¬ ¬ ¬ 𝜑) )
| 4:: | ⊢ ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑 →
𝜑))
| 5:3: | ⊢ ( ¬ ¬ 𝜑 ▶ (¬ ¬ 𝜑 → 𝜑) )
| 6:5,1: | ⊢ ( ¬ ¬ 𝜑 ▶ 𝜑 )
| qed:6: | ⊢ (¬ ¬ 𝜑 → 𝜑)
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ (¬ ¬
𝜑 → 𝜑) |
|
Theorem | con3ALTVD 41256 |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Theorem 7 of
Section 14 of [Margaris] p. 60 (which is con3 156). The same proof may
also be interpreted to be a Virtual Deduction Hilbert-style axiomatic
proof. It was completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. con3ALT2 40870 is con3ALTVD 41256 without
virtual deductions and was automatically derived from con3ALTVD 41256.
Step i of the User's Proof corresponds to step i of the Fitch-style proof.
1:: | ⊢ ( (𝜑 → 𝜓) ▶ (𝜑 → 𝜓) )
| 2:: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜑 )
| 3:: | ⊢ (¬ ¬ 𝜑 → 𝜑)
| 4:2: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜑 )
| 5:1,4: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜓 )
| 6:: | ⊢ (𝜓 → ¬ ¬ 𝜓)
| 7:6,5: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜓 )
| 8:7: | ⊢ ( (𝜑 → 𝜓) ▶ (¬ ¬ 𝜑 → ¬ ¬ 𝜓
) )
| 9:: | ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 →
¬ 𝜑))
| 10:8: | ⊢ ( (𝜑 → 𝜓) ▶ (¬ 𝜓 → ¬ 𝜑) )
| qed:10: | ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) |
|
20.36.9 Theorems proved using conjunction-form
Virtual Deduction
|
|
Theorem | elpwgdedVD 41257 |
Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived
from elpwg 4545. In form of VD deduction with 𝜑 and 𝜓 as
variable virtual hypothesis collections based on Mario Carneiro's
metavariable concept. elpwgded 40904 is elpwgdedVD 41257 using conventional
notation. (Contributed by Alan Sare, 23-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ ( 𝜑 ▶ 𝐴 ∈ V ) & ⊢ ( 𝜓 ▶ 𝐴 ⊆ 𝐵 )
⇒ ⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝐴 ∈ 𝒫 𝐵 ) |
|
Theorem | sspwimp 41258 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. For the biconditional, see
sspwb 5345. The proof sspwimp 41258, using conventional notation, was
translated from virtual deduction form, sspwimpVD 41259, using a
translation program. (Contributed by Alan Sare, 23-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
|
Theorem | sspwimpVD 41259 |
The following User's Proof is a Virtual Deduction proof (see wvd1 40909)
using conjunction-form virtual hypothesis collections. It was completed
manually, but has the potential to be completed automatically by a tools
program which would invoke Mel L. O'Cat's mmj2 and Norm Megill's
Metamath Proof Assistant.
sspwimp 41258 is sspwimpVD 41259 without virtual deductions and was derived
from sspwimpVD 41259. (Contributed by Alan Sare, 23-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
1:: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝐴 ⊆ 𝐵 )
| 2:: | ⊢ ( .............. 𝑥 ∈ 𝒫 𝐴
▶ 𝑥 ∈ 𝒫 𝐴 )
| 3:2: | ⊢ ( .............. 𝑥 ∈ 𝒫 𝐴
▶ 𝑥 ⊆ 𝐴 )
| 4:3,1: | ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ⊆ 𝐵 )
| 5:: | ⊢ 𝑥 ∈ V
| 6:4,5: | ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ∈ 𝒫 𝐵
)
| 7:6: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)
)
| 8:7: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈
𝒫 𝐵) )
| 9:8: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝒫 𝐴 ⊆ 𝒫 𝐵 )
| qed:9: | ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
|
|
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
|
Theorem | sspwimpcf 41260 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. sspwimpcf 41260, using
conventional notation, was translated from its virtual deduction form,
sspwimpcfVD 41261, using a translation program. (Contributed
by Alan Sare,
13-Jun-2015.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
|
Theorem | sspwimpcfVD 41261 |
The following User's Proof is a Virtual Deduction proof (see wvd1 40909)
using conjunction-form virtual hypothesis collections. It was completed
automatically by a tools program which would invokes Mel L. O'Cat's mmj2
and Norm Megill's Metamath Proof Assistant.
sspwimpcf 41260 is sspwimpcfVD 41261 without virtual deductions and was derived
from sspwimpcfVD 41261.
The version of completeusersproof.cmd used is capable of only generating
conjunction-form unification theorems, not unification deductions.
(Contributed by Alan Sare, 13-Jun-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
1:: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝐴 ⊆ 𝐵 )
| 2:: | ⊢ ( ........... 𝑥 ∈ 𝒫 𝐴
▶ 𝑥 ∈ 𝒫 𝐴 )
| 3:2: | ⊢ ( ........... 𝑥 ∈ 𝒫 𝐴
▶ 𝑥 ⊆ 𝐴 )
| 4:3,1: | ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ⊆ 𝐵 )
| 5:: | ⊢ 𝑥 ∈ V
| 6:4,5: | ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ∈ 𝒫 𝐵
)
| 7:6: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)
)
| 8:7: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈
𝒫 𝐵) )
| 9:8: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝒫 𝐴 ⊆ 𝒫 𝐵 )
| qed:9: | ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
|
|
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
|
Theorem | suctrALTcf 41262 |
The sucessor of a transitive class is transitive. suctrALTcf 41262, using
conventional notation, was translated from virtual deduction form,
suctrALTcfVD 41263, using a translation program. (Contributed
by Alan
Sare, 13-Jun-2015.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ (Tr 𝐴 → Tr suc 𝐴) |
|
Theorem | suctrALTcfVD 41263 |
The following User's Proof is a Virtual Deduction proof (see wvd1 40909)
using conjunction-form virtual hypothesis collections. The
conjunction-form version of completeusersproof.cmd. It allows the User
to avoid superflous virtual hypotheses. This proof was completed
automatically by a tools program which invokes Mel L. O'Cat's
mmj2 and Norm Megill's Metamath Proof Assistant. suctrALTcf 41262
is suctrALTcfVD 41263 without virtual deductions and was derived
automatically from suctrALTcfVD 41263. The version of
completeusersproof.cmd used is capable of only generating
conjunction-form unification theorems, not unification deductions.
(Contributed by Alan Sare, 13-Jun-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
1:: | ⊢ ( Tr 𝐴 ▶ Tr 𝐴 )
| 2:: | ⊢ ( ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) )
| 3:2: | ⊢ ( ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ 𝑧 ∈ 𝑦 )
| 4:: | ⊢ ( ...................................
....... 𝑦 ∈ 𝐴 ▶ 𝑦 ∈ 𝐴 )
| 5:1,3,4: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
, 𝑦 ∈ 𝐴 ) ▶ 𝑧 ∈ 𝐴 )
| 6:: | ⊢ 𝐴 ⊆ suc 𝐴
| 7:5,6: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
, 𝑦 ∈ 𝐴 ) ▶ 𝑧 ∈ suc 𝐴 )
| 8:7: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
) ▶ (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) )
| 9:: | ⊢ ( ...................................
...... 𝑦 = 𝐴 ▶ 𝑦 = 𝐴 )
| 10:3,9: | ⊢ ( ........ ( (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴), 𝑦 = 𝐴 ) ▶ 𝑧 ∈ 𝐴 )
| 11:10,6: | ⊢ ( ........ ( (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴), 𝑦 = 𝐴 ) ▶ 𝑧 ∈ suc 𝐴 )
| 12:11: | ⊢ ( .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) )
| 13:2: | ⊢ ( .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ 𝑦 ∈ suc 𝐴 )
| 14:13: | ⊢ ( .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) )
| 15:8,12,14: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
) ▶ 𝑧 ∈ suc 𝐴 )
| 16:15: | ⊢ ( Tr 𝐴 ▶ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) → 𝑧 ∈ suc 𝐴) )
| 17:16: | ⊢ ( Tr 𝐴 ▶ ∀𝑧∀𝑦((𝑧 ∈
𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) )
| 18:17: | ⊢ ( Tr 𝐴 ▶ Tr suc 𝐴 )
| qed:18: | ⊢ (Tr 𝐴 → Tr suc 𝐴)
|
|
⊢ (Tr 𝐴 → Tr suc 𝐴) |
|
20.36.10 Theorems with a VD proof in
conventional notation derived from a VD proof
|
|
Theorem | suctrALT3 41264 |
The successor of a transitive class is transitive. suctrALT3 41264 is the
completed proof in conventional notation of the Virtual Deduction proof
https://us.metamath.org/other/completeusersproof/suctralt3vd.html 41264.
It was completed manually. The potential for automated derivation from
the VD proof exists. See wvd1 40909 for a description of Virtual
Deduction.
Some sub-theorems of the proof were completed using a unification
deduction (e.g., the sub-theorem whose assertion is step 19 used
jaoded 40906). Unification deductions employ Mario
Carneiro's metavariable
concept. Some sub-theorems were completed using a unification theorem
(e.g., the sub-theorem whose assertion is step 24 used dftr2 5177) .
(Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (Tr 𝐴 → Tr suc 𝐴) |
|
Theorem | sspwimpALT 41265 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. sspwimpALT 41265 is the completed
proof in conventional notation of the Virtual Deduction proof
https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html 41265.
It was completed manually. The potential for automated derivation from
the VD proof exists. See wvd1 40909 for a description of Virtual
Deduction.
Some sub-theorems of the proof were completed using a unification
deduction (e.g., the sub-theorem whose assertion is step 9 used
elpwgded 40904). Unification deductions employ Mario
Carneiro's
metavariable concept. Some sub-theorems were completed using a
unification theorem (e.g., the sub-theorem whose assertion is step 5
used elpwi 4551). (Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
|
Theorem | unisnALT 41266 |
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
The User manually input on a mmj2 Proof Worksheet, without labels, all
steps of unisnALT 41266 except 1, 11, 15, 21, and 30. With
execution of the
mmj2 unification command, mmj2 could find labels for all steps except
for 2, 12, 16, 22, and 31 (and the then non-existing steps 1, 11, 15,
21, and 30). mmj2 could not find reference theorems for those five steps
because the hypothesis field of each of these steps was empty and none
of those steps unifies with a theorem in set.mm. Each of these five
steps is a semantic variation of a theorem in set.mm and is 2-step
provable. mmj2 does not have the ability to automatically generate the
semantic variation in set.mm of a theorem in a mmj2 Proof Worksheet
unless the theorem in the Proof Worksheet is labeled with a 1-hypothesis
deduction whose hypothesis is a theorem in set.mm which unifies with the
theorem in the Proof Worksheet. The stepprover.c program, which invokes
mmj2, has this capability. stepprover.c automatically generated steps 1,
11, 15, 21, and 30, labeled all steps, and generated the RPN proof of
unisnALT 41266. Roughly speaking, stepprover.c added to
the Proof
Worksheet a labeled duplicate step of each non-unifying theorem for each
label in a text file, labels.txt, containing a list of labels provided
by the User. Upon mmj2 unification, stepprover.c identified a label for
each of the five theorems which 2-step proves it. For unisnALT 41266, the
label list is a list of all 1-hypothesis propositional calculus
deductions in set.mm. stepproverp.c is the same as stepprover.c except
that it intermittently pauses during execution, allowing the User to
observe the changes to a text file caused by the execution of particular
statements of the program. (Contributed by Alan Sare, 19-Aug-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ 𝐴 ∈
V ⇒ ⊢ ∪
{𝐴} = 𝐴 |
|
20.36.11 Theorems with a proof in conventional
notation derived from a VD proof
Theorems with a proof in conventional notation automatically derived by
completeusersproof.c from a Virtual Deduction User's Proof.
|
|
Theorem | notnotrALT2 41267 |
Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102.
Proof derived by completeusersproof.c from User's Proof in
VirtualDeductionProofs.txt. (Contributed by Alan Sare, 11-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (¬ ¬
𝜑 → 𝜑) |
|
Theorem | sspwimpALT2 41268 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. Proof derived by
completeusersproof.c from User's Proof in VirtualDeductionProofs.txt.
The User's Proof in html format is displayed in
https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html.
(Contributed by Alan Sare, 11-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
|
Theorem | e2ebindALT 41269 |
Absorption of an existential quantifier of a double existential quantifier
of non-distinct variables. The proof is derived by completeusersproof.c
from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html
format is displayed in e2ebindVD 41252. (Contributed by Alan Sare,
11-Sep-2016.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
|
Theorem | ax6e2ndALT 41270* |
If at least two sets exist (dtru 5274) , then the same is true expressed
in an alternate form similar to the form of ax6e 2400.
The proof is
derived by completeusersproof.c from User's Proof in
VirtualDeductionProofs.txt. The User's Proof in html format is
displayed in ax6e2ndVD 41248. (Contributed by Alan Sare, 11-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
|
Theorem | ax6e2ndeqALT 41271* |
"At least two sets exist" expressed in the form of dtru 5274
is logically
equivalent to the same expressed in a form similar to ax6e 2400
if dtru 5274
is false implies 𝑢 = 𝑣. Proof derived by
completeusersproof.c from
User's Proof in VirtualDeductionProofs.txt. The User's Proof in html
format is displayed in ax6e2ndeqVD 41249. (Contributed by Alan Sare,
11-Sep-2016.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
|
Theorem | 2sb5ndALT 41272* |
Equivalence for double substitution 2sb5 2281 without distinct 𝑥,
𝑦 requirement. 2sb5nd 40900 is derived from 2sb5ndVD 41250. The proof is
derived by completeusersproof.c from User's Proof in
VirtualDeductionProofs.txt. The User's Proof in html format is
displayed in 2sb5ndVD 41250. (Contributed by Alan Sare, 19-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) |
|
Theorem | chordthmALT 41273* |
The intersecting chords theorem. If points A, B, C, and D lie on a
circle (with center Q, say), and the point P is on the interior of the
segments AB and CD, then the two products of lengths PA · PB and
PC · PD are equal. The Euclidean plane is identified with the
complex plane, and the fact that P is on AB and on CD is expressed by
the hypothesis that the angles APB and CPD are equal to π. The
result is proven by using chordthmlem5 25417 twice to show that PA
· PB and PC · PD both equal BQ
2
−
PQ
2
. This is similar to the proof of the
theorem given in Euclid's Elements, where it is Proposition
III.35.
Proven by David Moews on 28-Feb-2017 as chordthm 25418.
https://us.metamath.org/other/completeusersproof/chordthmaltvd.html 25418 is
a Virtual
Deduction User's Proof transcription of chordthm 25418. That VD User's
Proof was input into completeusersproof, automatically generating this
chordthmALT 41273 Metamath proof. (Contributed by Alan Sare,
19-Sep-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0})
↦ (ℑ‘(log‘(𝑦 / 𝑥)))) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝑃 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝑃)
& ⊢ (𝜑 → 𝐵 ≠ 𝑃)
& ⊢ (𝜑 → 𝐶 ≠ 𝑃)
& ⊢ (𝜑 → 𝐷 ≠ 𝑃)
& ⊢ (𝜑 → ((𝐴 − 𝑃)𝐹(𝐵 − 𝑃)) = π) & ⊢ (𝜑 → ((𝐶 − 𝑃)𝐹(𝐷 − 𝑃)) = π) & ⊢ (𝜑 → 𝑄 ∈ ℂ) & ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐵 − 𝑄))) & ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐶 − 𝑄))) & ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐷 − 𝑄))) ⇒ ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = ((abs‘(𝑃 − 𝐶)) · (abs‘(𝑃 − 𝐷)))) |
|
Theorem | isosctrlem1ALT 41274 |
Lemma for isosctr 25402. This proof was automatically derived by
completeusersproof from its Virtual Deduction proof counterpart
https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 25402.
As it is verified by the Metamath program, isosctrlem1ALT 41274 verifies
https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 41274.
(Contributed by Alan Sare, 22-Apr-2018.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ ((𝐴 ∈ ℂ ∧
(abs‘𝐴) = 1 ∧
¬ 1 = 𝐴) →
(ℑ‘(log‘(1 − 𝐴))) ≠ π) |
|
Theorem | iunconnlem2 41275* |
The indexed union of connected overlapping subspaces sharing a common
point is connected. This proof was automatically derived by
completeusersproof from its Virtual Deduction proof counterpart
https://us.metamath.org/other/completeusersproof/iunconlem2vd.html.
As it is verified by the Metamath program, iunconnlem2 41275 verifies
https://us.metamath.org/other/completeusersproof/iunconlem2vd.html 41275.
(Contributed by Alan Sare, 22-Apr-2018.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (𝜓 ↔ ((((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) ∧ (𝑢 ∩ ∪
𝑘 ∈ 𝐴 𝐵) ≠ ∅) ∧ (𝑣 ∩ ∪
𝑘 ∈ 𝐴 𝐵) ≠ ∅) ∧ (𝑢 ∩ 𝑣) ⊆ (𝑋 ∖ ∪ 𝑘 ∈ 𝐴 𝐵)) ∧ ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ (𝑢 ∪ 𝑣))) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝑋)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐽 ↾t 𝐵) ∈ Conn)
⇒ ⊢ (𝜑 → (𝐽 ↾t ∪ 𝑘 ∈ 𝐴 𝐵) ∈ Conn) |
|
Theorem | iunconnALT 41276* |
The indexed union of connected overlapping subspaces sharing a common
point is connected. This proof was automatically derived by
completeusersproof from its Virtual Deduction proof counterpart
https://us.metamath.org/other/completeusersproof/iunconaltvd.html.
As it is verified by the Metamath program, iunconnALT 41276 verifies
https://us.metamath.org/other/completeusersproof/iunconaltvd.html 41276.
(Contributed by Alan Sare, 22-Apr-2018.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝑋)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐽 ↾t 𝐵) ∈ Conn)
⇒ ⊢ (𝜑 → (𝐽 ↾t ∪ 𝑘 ∈ 𝐴 𝐵) ∈ Conn) |
|
Theorem | sineq0ALT 41277 |
A complex number whose sine is zero is an integer multiple of π.
The Virtual Deduction form of the proof is
https://us.metamath.org/other/completeusersproof/sineq0altvd.html.
The
Metamath form of the proof is sineq0ALT 41277. The Virtual Deduction proof
is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 25112.
The Virtual Deduction proof is verified by automatically transforming it
into the Metamath form of the proof using completeusersproof, which is
verified by the Metamath program. The proof of
https://us.metamath.org/other/completeusersproof/sineq0altro.html 25112 is a
form of the completed proof which preserves the Virtual Deduction proof's
step numbers and their ordering. (Contributed by Alan Sare, 13-Jun-2018.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ (𝐴 ∈ ℂ →
((sin‘𝐴) = 0 ↔
(𝐴 / π) ∈
ℤ)) |
|
20.37 Mathbox for Glauco
Siliprandi
|
|
20.37.1 Miscellanea
|
|
Theorem | evth2f 41278* |
A version of evth2 23567 using bound-variable hypotheses instead of
distinct
variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
⊢
Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑦𝐹
& ⊢ Ⅎ𝑥𝑋
& ⊢ Ⅎ𝑦𝑋
& ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (topGen‘ran
(,))
& ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑋 ≠ ∅)
⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
|
Theorem | elunif 41279* |
A version of eluni 4844 using bound-variable hypotheses instead of
distinct
variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
⊢
Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
|
Theorem | rzalf 41280 |
A version of rzal 4456 using bound-variable hypotheses instead of
distinct
variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
⊢ Ⅎ𝑥 𝐴 = ∅ ⇒ ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
|
Theorem | fvelrnbf 41281 |
A version of fvelrnb 6729 using bound-variable hypotheses instead of
distinct variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
⊢
Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵
& ⊢ Ⅎ𝑥𝐹 ⇒ ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)) |
|
Theorem | rfcnpre1 41282 |
If F is a continuous function with respect to the standard topology,
then the preimage A of the values greater than a given extended real B
is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
⊢
Ⅎ𝑥𝐵
& ⊢ Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑥𝜑
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑋 = ∪
𝐽 & ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)}
& ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
|
Theorem | ubelsupr 41283* |
If U belongs to A and U is an upper bound, then U is the sup of A.
(Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
⊢ ((𝐴 ⊆ ℝ ∧ 𝑈 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ 𝑈) → 𝑈 = sup(𝐴, ℝ, < )) |
|
Theorem | fsumcnf 41284* |
A finite sum of functions to complex numbers from a common topological
space is continuous, without disjoint var constraint x ph. The class
expression for B normally contains free variables k and x to index it.
(Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
⊢ 𝐾 =
(TopOpen‘ℂfld) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)) |
|
Theorem | mulltgt0 41285 |
The product of a negative and a positive number is negative. (Contributed
by Glauco Siliprandi, 20-Apr-2017.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 0 <
𝐵)) → (𝐴 · 𝐵) < 0) |
|
Theorem | rspcegf 41286 |
A version of rspcev 3626 using bound-variable hypotheses instead of
distinct variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
|
Theorem | rabexgf 41287 |
A version of rabexg 5237 using bound-variable hypotheses instead of
distinct variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
⊢
Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
|
Theorem | fcnre 41288 |
A function continuous with respect to the standard topology, is a real
mapping. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
⊢ 𝐾 = (topGen‘ran
(,))
& ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐶 = (𝐽 Cn 𝐾)
& ⊢ (𝜑 → 𝐹 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
|
Theorem | sumsnd 41289* |
A sum of a singleton is the term. The deduction version of sumsn 15104.
(Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
⊢ (𝜑 → Ⅎ𝑘𝐵)
& ⊢ Ⅎ𝑘𝜑
& ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝑀 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
|
Theorem | evthf 41290* |
A version of evth 23566 using bound-variable hypotheses instead of
distinct
variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
⊢
Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑦𝐹
& ⊢ Ⅎ𝑥𝑋
& ⊢ Ⅎ𝑦𝑋
& ⊢ Ⅎ𝑥𝜑
& ⊢ Ⅎ𝑦𝜑
& ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (topGen‘ran
(,))
& ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑋 ≠ ∅)
⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥)) |
|
Theorem | cnfex 41291 |
The class of continuous functions between two topologies is a set.
(Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V) |
|
Theorem | fnchoice 41292* |
For a finite set, a choice function exists, without using the axiom of
choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
⊢ (𝐴 ∈ Fin → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))) |
|
Theorem | refsumcn 41293* |
A finite sum of continuous real functions, from a common topological
space, is continuous. The class expression for B normally contains free
variables k and x to index it. See fsumcn 23481 for the analogous theorem
on continuous complex functions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)) |
|
Theorem | rfcnpre2 41294 |
If 𝐹 is a continuous function with
respect to the standard topology,
then the preimage A of the values smaller than a given extended real
𝐵, is an open set. (Contributed by
Glauco Siliprandi,
20-Apr-2017.)
|
⊢
Ⅎ𝑥𝐵
& ⊢ Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑥𝜑
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑋 = ∪
𝐽 & ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵}
& ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
|
Theorem | cncmpmax 41295* |
When the hypothesis for the extreme value theorem hold, then the sup of
the range of the function belongs to the range, it is real and it an
upper bound of the range. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
⊢ 𝑇 = ∪
𝐽 & ⊢ 𝐾 = (topGen‘ran
(,))
& ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑇 ≠ ∅)
⇒ ⊢ (𝜑 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈
ℝ ∧ ∀𝑡
∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ))) |
|
Theorem | rfcnpre3 41296* |
If F is a continuous function with respect to the standard topology,
then the preimage A of the values greater than or equal to a given real
B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
⊢
Ⅎ𝑡𝐹
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑇 = ∪
𝐽 & ⊢ 𝐴 = {𝑡 ∈ 𝑇 ∣ 𝐵 ≤ (𝐹‘𝑡)}
& ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) |
|
Theorem | rfcnpre4 41297* |
If F is a continuous function with respect to the standard topology,
then the preimage A of the values less than or equal to a given real B
is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
⊢
Ⅎ𝑡𝐹
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑇 = ∪
𝐽 & ⊢ 𝐴 = {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ 𝐵}
& ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) |
|
Theorem | sumpair 41298* |
Sum of two distinct complex values. The class expression for 𝐴 and
𝐵 normally contain free variable 𝑘 to
index it. (Contributed by
Glauco Siliprandi, 20-Apr-2017.)
|
⊢ (𝜑 → Ⅎ𝑘𝐷)
& ⊢ (𝜑 → Ⅎ𝑘𝐸)
& ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊)
& ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐸 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
& ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸)) |
|
Theorem | rfcnnnub 41299* |
Given a real continuous function 𝐹 defined on a compact topological
space, there is always a positive integer that is a strict upper bound
of its range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
⊢
Ⅎ𝑡𝐹
& ⊢ Ⅎ𝑡𝜑
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ 𝑇 = ∪
𝐽 & ⊢ (𝜑 → 𝑇 ≠ ∅) & ⊢ 𝐶 = (𝐽 Cn 𝐾)
& ⊢ (𝜑 → 𝐹 ∈ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑛) |
|
Theorem | refsum2cnlem1 41300* |
This is the core Lemma for refsum2cn 41301: the sum of two continuous real
functions (from a common topological space) is continuous. (Contributed
by Glauco Siliprandi, 20-Apr-2017.)
|
⊢
Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑥𝐺
& ⊢ Ⅎ𝑥𝜑
& ⊢ 𝐴 = (𝑘 ∈ {1, 2} ↦ if(𝑘 = 1, 𝐹, 𝐺)) & ⊢ 𝐾 = (topGen‘ran
(,))
& ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) ∈ (𝐽 Cn 𝐾)) |