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Theorem onfrALT 44243
Description: The membership relation is foundational on the class of ordinal numbers. onfrALT 44243 is an alternate proof of onfr 6417. onfrALTVD 44585 is the Virtual Deduction proof from which onfrALT 44243 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6417 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 44585. 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 5669 . 2 ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
2 simpr 483 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
3 n0 4349 . . . 4 (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎)
4 onfrALTlem1 44242 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
54expd 414 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → ((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
6 onfrALTlem2 44240 . . . . . . 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 1929 . . . 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 1794 1 E Fr On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1534  wex 1774  wne 2930  wrex 3060  cin 3946  wss 3947  c0 4325   E cep 5587   Fr wfr 5636  Oncon0 6378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2366  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-tr 5273  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5639  df-we 5641  df-ord 6381  df-on 6382
This theorem is referenced by: (None)
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