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Theorem onfrALT 28572
 Description: The epsilon relation is foundational on the class of ordinal numbers. onfrALT 28572 is an alternate proof of onfr 4612. onfrALTVD 28940 is the Virtual Deduction proof from which onfrALT 28572 is derived. The Virtual Deduction proof mirrors the working proof of onfr 4612 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 28940. 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

Proof of Theorem onfrALT
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfepfr 4559 . 2
2 simpr 448 . . 3
3 n0 3629 . . . 4
4 onfrALTlem1 28571 . . . . . . 7
54exp3a 426 . . . . . 6
6 onfrALTlem2 28569 . . . . . . 7
76exp3a 426 . . . . . 6
8 pm2.61 165 . . . . . 6
95, 7, 8ee22 1371 . . . . 5
109exlimdv 1646 . . . 4
113, 10syl5bi 209 . . 3
122, 11mpd 15 . 2
131, 12mpgbir 1559 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359  wex 1550   wceq 1652   wne 2598  wrex 2698   cin 3311   wss 3312  c0 3620   cep 4484   wfr 4530  con0 4573 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
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