Proof of Theorem tfsconcat0b
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nnon 7894 | . . . . 5
⊢ (𝐷 ∈ ω → 𝐷 ∈ On) | 
| 2 | 1 | anim2i 617 | . . . 4
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (𝐶 ∈ On ∧ 𝐷 ∈ On)) | 
| 3 | 2 | anim2i 617 | . . 3
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On))) | 
| 4 |  | tfsconcat.op | . . . 4
⊢  + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) | 
| 5 | 4 | tfsconcat0i 43363 | . . 3
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 = ∅ → (𝐴 + 𝐵) = 𝐵)) | 
| 6 | 3, 5 | syl 17 | . 2
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ → (𝐴 + 𝐵) = 𝐵)) | 
| 7 |  | dmeq 5913 | . . 3
⊢ ((𝐴 + 𝐵) = 𝐵 → dom (𝐴 + 𝐵) = dom 𝐵) | 
| 8 |  | nna0r 8648 | . . . . . . . . 9
⊢ (𝐷 ∈ ω → (∅
+o 𝐷) = 𝐷) | 
| 9 | 8 | adantl 481 | . . . . . . . 8
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅
+o 𝐷) = 𝐷) | 
| 10 | 9 | eqeq2d 2747 | . . . . . . 7
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = (∅ +o 𝐷) ↔ (𝐶 +o 𝐷) = 𝐷)) | 
| 11 |  | eqcom 2743 | . . . . . . 7
⊢ ((𝐶 +o 𝐷) = (∅ +o 𝐷) ↔ (∅ +o 𝐷) = (𝐶 +o 𝐷)) | 
| 12 | 10, 11 | bitr3di 286 | . . . . . 6
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = 𝐷 ↔ (∅ +o 𝐷) = (𝐶 +o 𝐷))) | 
| 13 |  | on0eln0 6439 | . . . . . . . . . 10
⊢ (𝐶 ∈ On → (∅
∈ 𝐶 ↔ 𝐶 ≠ ∅)) | 
| 14 | 13 | adantr 480 | . . . . . . . . 9
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅
∈ 𝐶 ↔ 𝐶 ≠ ∅)) | 
| 15 |  | df-ne 2940 | . . . . . . . . 9
⊢ (𝐶 ≠ ∅ ↔ ¬ 𝐶 = ∅) | 
| 16 | 14, 15 | bitr2di 288 | . . . . . . . 8
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (¬
𝐶 = ∅ ↔ ∅
∈ 𝐶)) | 
| 17 |  | peano1 7911 | . . . . . . . . . . . . . . 15
⊢ ∅
∈ ω | 
| 18 |  | nnaordr 8659 | . . . . . . . . . . . . . . 15
⊢ ((∅
∈ ω ∧ 𝐶
∈ ω ∧ 𝐷
∈ ω) → (∅ ∈ 𝐶 ↔ (∅ +o 𝐷) ∈ (𝐶 +o 𝐷))) | 
| 19 | 17, 18 | mp3an1 1449 | . . . . . . . . . . . . . 14
⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅
∈ 𝐶 ↔ (∅
+o 𝐷) ∈
(𝐶 +o 𝐷))) | 
| 20 | 19 | biimpd 229 | . . . . . . . . . . . . 13
⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ ω) → (∅
∈ 𝐶 → (∅
+o 𝐷) ∈
(𝐶 +o 𝐷))) | 
| 21 | 20 | ex 412 | . . . . . . . . . . . 12
⊢ (𝐶 ∈ ω → (𝐷 ∈ ω → (∅
∈ 𝐶 → (∅
+o 𝐷) ∈
(𝐶 +o 𝐷)))) | 
| 22 | 21 | a1i 11 | . . . . . . . . . . 11
⊢ (𝐶 ∈ On → (𝐶 ∈ ω → (𝐷 ∈ ω → (∅
∈ 𝐶 → (∅
+o 𝐷) ∈
(𝐶 +o 𝐷))))) | 
| 23 |  | simpr 484 | . . . . . . . . . . . . . . . 16
⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω
⊆ 𝐶) → ω
⊆ 𝐶) | 
| 24 |  | oaword1 8591 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝐷)) | 
| 25 | 2, 24 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → 𝐶 ⊆ (𝐶 +o 𝐷)) | 
| 26 | 25 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω
⊆ 𝐶) → 𝐶 ⊆ (𝐶 +o 𝐷)) | 
| 27 | 23, 26 | sstrd 3993 | . . . . . . . . . . . . . . 15
⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω
⊆ 𝐶) → ω
⊆ (𝐶 +o
𝐷)) | 
| 28 |  | id 22 | . . . . . . . . . . . . . . . . 17
⊢ (𝐷 ∈ ω → 𝐷 ∈
ω) | 
| 29 | 8, 28 | eqeltrd 2840 | . . . . . . . . . . . . . . . 16
⊢ (𝐷 ∈ ω → (∅
+o 𝐷) ∈
ω) | 
| 30 | 29 | ad2antlr 727 | . . . . . . . . . . . . . . 15
⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω
⊆ 𝐶) → (∅
+o 𝐷) ∈
ω) | 
| 31 | 27, 30 | sseldd 3983 | . . . . . . . . . . . . . 14
⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω
⊆ 𝐶) → (∅
+o 𝐷) ∈
(𝐶 +o 𝐷)) | 
| 32 | 31 | a1d 25 | . . . . . . . . . . . . 13
⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ ω) ∧ ω
⊆ 𝐶) → (∅
∈ 𝐶 → (∅
+o 𝐷) ∈
(𝐶 +o 𝐷))) | 
| 33 | 32 | exp31 419 | . . . . . . . . . . . 12
⊢ (𝐶 ∈ On → (𝐷 ∈ ω → (ω
⊆ 𝐶 → (∅
∈ 𝐶 → (∅
+o 𝐷) ∈
(𝐶 +o 𝐷))))) | 
| 34 | 33 | com23 86 | . . . . . . . . . . 11
⊢ (𝐶 ∈ On → (ω
⊆ 𝐶 → (𝐷 ∈ ω → (∅
∈ 𝐶 → (∅
+o 𝐷) ∈
(𝐶 +o 𝐷))))) | 
| 35 |  | eloni 6393 | . . . . . . . . . . . 12
⊢ (𝐶 ∈ On → Ord 𝐶) | 
| 36 |  | ordom 7898 | . . . . . . . . . . . 12
⊢ Ord
ω | 
| 37 |  | ordtri2or 6481 | . . . . . . . . . . . 12
⊢ ((Ord
𝐶 ∧ Ord ω) →
(𝐶 ∈ ω ∨
ω ⊆ 𝐶)) | 
| 38 | 35, 36, 37 | sylancl 586 | . . . . . . . . . . 11
⊢ (𝐶 ∈ On → (𝐶 ∈ ω ∨ ω
⊆ 𝐶)) | 
| 39 | 22, 34, 38 | mpjaod 860 | . . . . . . . . . 10
⊢ (𝐶 ∈ On → (𝐷 ∈ ω → (∅
∈ 𝐶 → (∅
+o 𝐷) ∈
(𝐶 +o 𝐷)))) | 
| 40 | 39 | imp 406 | . . . . . . . . 9
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅
∈ 𝐶 → (∅
+o 𝐷) ∈
(𝐶 +o 𝐷))) | 
| 41 |  | elneq 9639 | . . . . . . . . . 10
⊢ ((∅
+o 𝐷) ∈
(𝐶 +o 𝐷) → (∅ +o
𝐷) ≠ (𝐶 +o 𝐷)) | 
| 42 | 41 | neneqd 2944 | . . . . . . . . 9
⊢ ((∅
+o 𝐷) ∈
(𝐶 +o 𝐷) → ¬ (∅
+o 𝐷) = (𝐶 +o 𝐷)) | 
| 43 | 40, 42 | syl6 35 | . . . . . . . 8
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (∅
∈ 𝐶 → ¬
(∅ +o 𝐷) =
(𝐶 +o 𝐷))) | 
| 44 | 16, 43 | sylbid 240 | . . . . . . 7
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → (¬
𝐶 = ∅ → ¬
(∅ +o 𝐷) =
(𝐶 +o 𝐷))) | 
| 45 | 44 | con4d 115 | . . . . . 6
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) →
((∅ +o 𝐷)
= (𝐶 +o 𝐷) → 𝐶 = ∅)) | 
| 46 | 12, 45 | sylbid 240 | . . . . 5
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ ω) → ((𝐶 +o 𝐷) = 𝐷 → 𝐶 = ∅)) | 
| 47 | 46 | adantl 481 | . . . 4
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → ((𝐶 +o 𝐷) = 𝐷 → 𝐶 = ∅)) | 
| 48 | 4 | tfsconcatfn 43356 | . . . . . . 7
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷)) | 
| 49 | 3, 48 | syl 17 | . . . . . 6
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 + 𝐵) Fn (𝐶 +o 𝐷)) | 
| 50 | 49 | fndmd 6672 | . . . . 5
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom (𝐴 + 𝐵) = (𝐶 +o 𝐷)) | 
| 51 |  | fndm 6670 | . . . . . 6
⊢ (𝐵 Fn 𝐷 → dom 𝐵 = 𝐷) | 
| 52 | 51 | ad2antlr 727 | . . . . 5
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → dom 𝐵 = 𝐷) | 
| 53 | 50, 52 | eqeq12d 2752 | . . . 4
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (dom (𝐴 + 𝐵) = dom 𝐵 ↔ (𝐶 +o 𝐷) = 𝐷)) | 
| 54 |  | fnrel 6669 | . . . . . . . 8
⊢ (𝐴 Fn 𝐶 → Rel 𝐴) | 
| 55 |  | reldm0 5937 | . . . . . . . 8
⊢ (Rel
𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) | 
| 56 | 54, 55 | syl 17 | . . . . . . 7
⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) | 
| 57 |  | fndm 6670 | . . . . . . . 8
⊢ (𝐴 Fn 𝐶 → dom 𝐴 = 𝐶) | 
| 58 | 57 | eqeq1d 2738 | . . . . . . 7
⊢ (𝐴 Fn 𝐶 → (dom 𝐴 = ∅ ↔ 𝐶 = ∅)) | 
| 59 | 56, 58 | bitrd 279 | . . . . . 6
⊢ (𝐴 Fn 𝐶 → (𝐴 = ∅ ↔ 𝐶 = ∅)) | 
| 60 | 59 | adantr 480 | . . . . 5
⊢ ((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) → (𝐴 = ∅ ↔ 𝐶 = ∅)) | 
| 61 | 60 | adantr 480 | . . . 4
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ ↔ 𝐶 = ∅)) | 
| 62 | 47, 53, 61 | 3imtr4d 294 | . . 3
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (dom (𝐴 + 𝐵) = dom 𝐵 → 𝐴 = ∅)) | 
| 63 | 7, 62 | syl5 34 | . 2
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → ((𝐴 + 𝐵) = 𝐵 → 𝐴 = ∅)) | 
| 64 | 6, 63 | impbid 212 | 1
⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ ↔ (𝐴 + 𝐵) = 𝐵)) |