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Theorem onfrALT 39359
Description: The epsilon relation is foundational on the class of ordinal numbers. onfrALT 39359 is an alternate proof of onfr 5947. onfrALTVD 39711 is the Virtual Deduction proof from which onfrALT 39359 is derived. The Virtual Deduction proof mirrors the working proof of onfr 5947 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 39711. 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 5262 . 2 ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
2 simpr 477 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
3 n0 4095 . . . 4 (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎)
4 onfrALTlem1 39358 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
54expd 404 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → ((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
6 onfrALTlem2 39356 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
76expd 404 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → (¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
8 pm2.61 183 . . . . . 6 (((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ((¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
95, 7, 8syl6c 70 . . . . 5 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
109exlimdv 2028 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
113, 10syl5bi 233 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅))
122, 11mpd 15 . 2 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)
131, 12mpgbir 1894 1 E Fr On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1652  wex 1874  wne 2937  wrex 3056  cin 3731  wss 3732  c0 4079   E cep 5189   Fr wfr 5233  Oncon0 5908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-tr 4912  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-ord 5911  df-on 5912
This theorem is referenced by: (None)
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