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Theorem onfrALT 27330
Description: The epsilon relation is foundational on the class of ordinal numbers. onfrALT 27330 is an alternate proof of onfr 4368. onfrALTVD 27680 is the Virtual Deduction proof from which onfrALT 27330 is derived. The Virtual Deduction proof mirrors the working proof of onfr 4368 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 27680. 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
StepHypRef Expression
1 dfepfr 4315 . 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 3406 . . . 4  |-  ( a  =/=  (/)  <->  E. x  x  e.  a )
4 onfrALTlem1 27329 . . . . . . 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 27327 . . . . . . 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 pm2.61 165 . . . . . 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 1358 . . . . 5  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
x  e.  a  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) ) )
109exlimdv 1933 . . . 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 16 . 2  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )
131, 12mpgbir 1544 1  |-  _E  Fr  On
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360   E.wex 1537    = wceq 1619    =/= wne 2419   E.wrex 2517    i^i cin 3093    C_ wss 3094   (/)c0 3397    _E cep 4240    Fr wfr 4286   Oncon0 4329
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-tr 4054  df-eprel 4242  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333
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