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Theorem onfrALT 28697
Description: The epsilon relation is foundational on the class of ordinal numbers. onfrALT 28697 is an alternate proof of onfr 4622. onfrALTVD 29065 is the Virtual Deduction proof from which onfrALT 28697 is derived. The Virtual Deduction proof mirrors the working proof of onfr 4622 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 29065. 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 4569 . 2  |-  (  _E  Fr  On  <->  A. a
( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) ) )
2 simpr 449 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  =/=  (/) )
3 n0 3639 . . . 4  |-  ( a  =/=  (/)  <->  E. x  x  e.  a )
4 onfrALTlem1 28696 . . . . . . 7  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) )
54exp3a 427 . . . . . 6  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
x  e.  a  -> 
( ( a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ) )
6 onfrALTlem2 28694 . . . . . . 7  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) )
76exp3a 427 . . . . . 6  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
x  e.  a  -> 
( -.  ( a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ) )
8 pm2.61 166 . . . . . 6  |-  ( ( ( a  i^i  x
)  =  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  ( ( -.  ( a  i^i  x
)  =  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) )
95, 7, 8ee22 1372 . . . . 5  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
x  e.  a  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) ) )
109exlimdv 1647 . . . 4  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  ( E. x  x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) )
113, 10syl5bi 210 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
a  =/=  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) )
122, 11mpd 15 . 2  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )
131, 12mpgbir 1560 1  |-  _E  Fr  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    =/= wne 2601   E.wrex 2708    i^i cin 3321    C_ wss 3322   (/)c0 3630    _E cep 4494    Fr wfr 4540   Oncon0 4583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587
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