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Theorem onfrALT 43612
Description: The membership relation is foundational on the class of ordinal numbers. onfrALT 43612 is an alternate proof of onfr 6402. onfrALTVD 43954 is the Virtual Deduction proof from which onfrALT 43612 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6402 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 43954. 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.)
Assertion
Ref Expression
onfrALT E Fr On

Proof of Theorem onfrALT
Dummy variables 𝑥 𝑎 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfepfr 5660 . 2 ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
2 simpr 483 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
3 n0 4345 . . . 4 (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎)
4 onfrALTlem1 43611 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
54expd 414 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → ((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
6 onfrALTlem2 43609 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
76expd 414 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → (¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
8 pm2.61 191 . . . . . 6 (((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ((¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
95, 7, 8syl6c 70 . . . . 5 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
109exlimdv 1934 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
113, 10biimtrid 241 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅))
122, 11mpd 15 . 2 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)
131, 12mpgbir 1799 1 E Fr On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1539  wex 1779  wne 2938  wrex 3068  cin 3946  wss 3947  c0 4321   E cep 5578   Fr wfr 5627  Oncon0 6363
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-13 2369  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6366  df-on 6367
This theorem is referenced by: (None)
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