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Theorem onfrALT 27104
Description: The epsilon relation is foundational on the class of ordinal numbers. onfrALT 27104 is an alternate proof of onfr 4324. onfrALTVD 27454 is the Virtual Deduction proof from which onfrALT 27104 is derived. The Virtual Deduction proof mirrors the working proof of onfr 4324 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 27454. 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 4271 . 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 3371 . . . 4  |-  ( a  =/=  (/)  <->  E. x  x  e.  a )
4 onfrALTlem1 27103 . . . . . . 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 27101 . . . . . . 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 1932 . . . 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 2412   E.wrex 2510    i^i cin 3077    C_ wss 3078   (/)c0 3362    _E cep 4196    Fr wfr 4242   Oncon0 4285
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289
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