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Theorem onfrALT 37785
 Description: The epsilon relation is foundational on the class of ordinal numbers. onfrALT 37785 is an alternate proof of onfr 5680. onfrALTVD 38149 is the Virtual Deduction proof from which onfrALT 37785 is derived. The Virtual Deduction proof mirrors the working proof of onfr 5680 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 38149. 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 5023 . 2 ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
2 simpr 476 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
3 n0 3890 . . . 4 (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎)
4 onfrALTlem1 37784 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
54expd 451 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → ((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
6 onfrALTlem2 37782 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
76expd 451 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → (¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
8 pm2.61 182 . . . . . 6 (((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ((¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
95, 7, 8syl6c 68 . . . . 5 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
109exlimdv 1848 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
113, 10syl5bi 231 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅))
122, 11mpd 15 . 2 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)
131, 12mpgbir 1717 1 E Fr On
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ≠ wne 2780  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   E cep 4947   Fr wfr 4994  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644 This theorem is referenced by: (None)
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