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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 43300 is an alternate proof of onfr 6403. onfrALTVD 43642 is the Virtual Deduction proof from which onfrALT 43300 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6403 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 43642. 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 5661 | . 2 ⊢ ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | |
2 | simpr 485 | . . 3 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅) | |
3 | n0 4346 | . . . 4 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎) | |
4 | onfrALTlem1 43299 | . . . . . . 7 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | |
5 | 4 | expd 416 | . . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥 ∈ 𝑎 → ((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))) |
6 | onfrALTlem2 43297 | . . . . . . 7 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | |
7 | 6 | expd 416 | . . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥 ∈ 𝑎 → (¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))) |
8 | pm2.61 191 | . . . . . 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 1936 | . . . 4 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) |
11 | 3, 10 | biimtrid 241 | . . 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 1801 | 1 ⊢ E Fr On |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ≠ wne 2940 ∃wrex 3070 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 E cep 5579 Fr wfr 5628 Oncon0 6364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2371 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-ord 6367 df-on 6368 |
This theorem is referenced by: (None) |
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