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Theorem onfrALT 44547
Description: The membership relation is foundational on the class of ordinal numbers. onfrALT 44547 is an alternate proof of onfr 6425. onfrALTVD 44889 is the Virtual Deduction proof from which onfrALT 44547 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6425 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 44889. 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 5673 . 2 ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
2 simpr 484 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
3 n0 4359 . . . 4 (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥𝑎)
4 onfrALTlem1 44546 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
54expd 415 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → ((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
6 onfrALTlem2 44544 . . . . . . 7 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
76expd 415 . . . . . 6 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → (¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅)))
8 pm2.61 192 . . . . . 6 (((𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ((¬ (𝑎𝑥) = ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))
95, 7, 8syl6c 70 . . . . 5 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
109exlimdv 1931 . . . 4 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (∃𝑥 𝑥𝑎 → ∃𝑦𝑎 (𝑎𝑦) = ∅))
113, 10biimtrid 242 . . 3 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑎 ≠ ∅ → ∃𝑦𝑎 (𝑎𝑦) = ∅))
122, 11mpd 15 . 2 ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅)
131, 12mpgbir 1796 1 E Fr On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wex 1776  wne 2938  wrex 3068  cin 3962  wss 3963  c0 4339   E cep 5588   Fr wfr 5638  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by: (None)
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