| Step | Hyp | Ref
| Expression |
| 1 | | simpr 484 |
. . . . . . . . . . . 12
⊢ (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ (𝑎 ∩ 𝑦)) |
| 2 | 1 | 2a1i 12 |
. . . . . . . . . . 11
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ (𝑎 ∩ 𝑦)))) |
| 3 | | inss2 4238 |
. . . . . . . . . . . 12
⊢ (𝑎 ∩ 𝑦) ⊆ 𝑦 |
| 4 | 3 | sseli 3979 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ 𝑦) |
| 5 | 2, 4 | syl8 76 |
. . . . . . . . . 10
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ 𝑦))) |
| 6 | | inss1 4237 |
. . . . . . . . . . . . 13
⊢ (𝑎 ∩ 𝑦) ⊆ 𝑎 |
| 7 | 6 | sseli 3979 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ 𝑎) |
| 8 | 2, 7 | syl8 76 |
. . . . . . . . . . 11
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ 𝑎))) |
| 9 | | simpl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ On) |
| 10 | | simpl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → 𝑥 ∈ 𝑎) |
| 11 | | ssel 3977 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ⊆ On → (𝑥 ∈ 𝑎 → 𝑥 ∈ On)) |
| 12 | 9, 10, 11 | syl2im 40 |
. . . . . . . . . . . . . 14
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → 𝑥 ∈ On)) |
| 13 | | eloni 6394 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ On → Ord 𝑥) |
| 14 | 12, 13 | syl6 35 |
. . . . . . . . . . . . 13
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → Ord 𝑥)) |
| 15 | | ordtr 6398 |
. . . . . . . . . . . . 13
⊢ (Ord
𝑥 → Tr 𝑥) |
| 16 | 14, 15 | syl6 35 |
. . . . . . . . . . . 12
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → Tr 𝑥)) |
| 17 | | simpll 767 |
. . . . . . . . . . . . . 14
⊢ (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑦 ∈ (𝑎 ∩ 𝑥)) |
| 18 | 17 | 2a1i 12 |
. . . . . . . . . . . . 13
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑦 ∈ (𝑎 ∩ 𝑥)))) |
| 19 | | inss2 4238 |
. . . . . . . . . . . . . 14
⊢ (𝑎 ∩ 𝑥) ⊆ 𝑥 |
| 20 | 19 | sseli 3979 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (𝑎 ∩ 𝑥) → 𝑦 ∈ 𝑥) |
| 21 | 18, 20 | syl8 76 |
. . . . . . . . . . . 12
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑦 ∈ 𝑥))) |
| 22 | | trel 5268 |
. . . . . . . . . . . . 13
⊢ (Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥)) |
| 23 | 22 | expcomd 416 |
. . . . . . . . . . . 12
⊢ (Tr 𝑥 → (𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥))) |
| 24 | 16, 21, 5, 23 | ee233 44539 |
. . . . . . . . . . 11
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ 𝑥))) |
| 25 | | elin 3967 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ (𝑎 ∩ 𝑥) ↔ (𝑧 ∈ 𝑎 ∧ 𝑧 ∈ 𝑥)) |
| 26 | 25 | simplbi2 500 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ 𝑎 → (𝑧 ∈ 𝑥 → 𝑧 ∈ (𝑎 ∩ 𝑥))) |
| 27 | 8, 24, 26 | ee33 44541 |
. . . . . . . . . 10
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ (𝑎 ∩ 𝑥)))) |
| 28 | | elin 3967 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦) ↔ (𝑧 ∈ (𝑎 ∩ 𝑥) ∧ 𝑧 ∈ 𝑦)) |
| 29 | 28 | simplbi2com 502 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝑦 → (𝑧 ∈ (𝑎 ∩ 𝑥) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦))) |
| 30 | 5, 27, 29 | ee33 44541 |
. . . . . . . . 9
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)))) |
| 31 | 30 | exp4a 431 |
. . . . . . . 8
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦))))) |
| 32 | 31 | ggen31 44565 |
. . . . . . 7
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → ∀𝑧(𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦))))) |
| 33 | | df-ss 3968 |
. . . . . . . 8
⊢ ((𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦) ↔ ∀𝑧(𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦))) |
| 34 | 33 | biimpri 228 |
. . . . . . 7
⊢
(∀𝑧(𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)) → (𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦)) |
| 35 | 32, 34 | syl8 76 |
. . . . . 6
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦)))) |
| 36 | | simpr 484 |
. . . . . . 7
⊢ ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) |
| 37 | 36 | 2a1i 12 |
. . . . . 6
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))) |
| 38 | | sseq0 4403 |
. . . . . . 7
⊢ (((𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑎 ∩ 𝑦) = ∅) |
| 39 | 38 | ex 412 |
. . . . . 6
⊢ ((𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦) → (((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅ → (𝑎 ∩ 𝑦) = ∅)) |
| 40 | 35, 37, 39 | ee33 44541 |
. . . . 5
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑎 ∩ 𝑦) = ∅))) |
| 41 | | simpl 482 |
. . . . . . 7
⊢ ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → 𝑦 ∈ (𝑎 ∩ 𝑥)) |
| 42 | 41 | 2a1i 12 |
. . . . . 6
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → 𝑦 ∈ (𝑎 ∩ 𝑥)))) |
| 43 | | inss1 4237 |
. . . . . . 7
⊢ (𝑎 ∩ 𝑥) ⊆ 𝑎 |
| 44 | 43 | sseli 3979 |
. . . . . 6
⊢ (𝑦 ∈ (𝑎 ∩ 𝑥) → 𝑦 ∈ 𝑎) |
| 45 | 42, 44 | syl8 76 |
. . . . 5
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → 𝑦 ∈ 𝑎))) |
| 46 | | pm3.21 471 |
. . . . 5
⊢ ((𝑎 ∩ 𝑦) = ∅ → (𝑦 ∈ 𝑎 → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))) |
| 47 | 40, 45, 46 | ee33 44541 |
. . . 4
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)))) |
| 48 | 47 | alrimdv 1929 |
. . 3
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∀𝑦((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)))) |
| 49 | | onfrALTlem3 44564 |
. . . 4
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)) |
| 50 | | df-rex 3071 |
. . . 4
⊢
(∃𝑦 ∈
(𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)) |
| 51 | 49, 50 | imbitrdi 251 |
. . 3
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))) |
| 52 | | exim 1834 |
. . 3
⊢
(∀𝑦((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) → (∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))) |
| 53 | 48, 51, 52 | syl6c 70 |
. 2
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))) |
| 54 | | df-rex 3071 |
. 2
⊢
(∃𝑦 ∈
𝑎 (𝑎 ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
| 55 | 53, 54 | imbitrrdi 252 |
1
⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) |