| Mathbox for Alan Sare |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > onfrALT | Structured version Visualization version GIF version | ||
| Description: The membership relation is foundational on the class of ordinal numbers. onfrALT 44569 is an alternate proof of onfr 6423. onfrALTVD 44911 is the Virtual Deduction proof from which onfrALT 44569 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6423 which is the main part of the proof of Theorem 7.12 of the first edition of TakeutiZaring. The proof of the corresponding Proposition 7.12 of [TakeutiZaring] p. 38 (second edition) does not contain the working proof equivalent of onfrALTVD 44911. This theorem does not rely on the Axiom of Regularity. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| onfrALT | ⊢ E Fr On |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfepfr 5669 | . 2 ⊢ ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | |
| 2 | simpr 484 | . . 3 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅) | |
| 3 | n0 4353 | . . . 4 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎) | |
| 4 | onfrALTlem1 44568 | . . . . . . 7 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | |
| 5 | 4 | expd 415 | . . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥 ∈ 𝑎 → ((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))) |
| 6 | onfrALTlem2 44566 | . . . . . . 7 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | |
| 7 | 6 | expd 415 | . . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥 ∈ 𝑎 → (¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))) |
| 8 | pm2.61 192 | . . . . . 6 ⊢ (((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → ((¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | |
| 9 | 5, 7, 8 | syl6c 70 | . . . . 5 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) |
| 10 | 9 | exlimdv 1933 | . . . 4 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) |
| 11 | 3, 10 | biimtrid 242 | . . 3 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) |
| 12 | 2, 11 | mpd 15 | . 2 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) |
| 13 | 1, 12 | mpgbir 1799 | 1 ⊢ E Fr On |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ≠ wne 2940 ∃wrex 3070 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 E cep 5583 Fr wfr 5634 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |