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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 40890 is an alternate proof of onfr 6232. onfrALTVD 41232 is the Virtual Deduction proof from which onfrALT 40890 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6232 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 41232. 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 5542 | . 2 ⊢ ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | |
2 | simpr 487 | . . 3 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅) | |
3 | n0 4312 | . . . 4 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎) | |
4 | onfrALTlem1 40889 | . . . . . . 7 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | |
5 | 4 | expd 418 | . . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥 ∈ 𝑎 → ((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))) |
6 | onfrALTlem2 40887 | . . . . . . 7 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) | |
7 | 6 | expd 418 | . . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥 ∈ 𝑎 → (¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))) |
8 | pm2.61 194 | . . . . . 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 1934 | . . . 4 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)) |
11 | 3, 10 | syl5bi 244 | . . 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 1800 | 1 ⊢ E Fr On |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ≠ wne 3018 ∃wrex 3141 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 E cep 5466 Fr wfr 5513 Oncon0 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-on 6197 |
This theorem is referenced by: (None) |
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