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Theorem onfrALT 28061
Description: The epsilon relation is foundational on the class of ordinal numbers. onfrALT 28061 is an alternate proof of onfr 4534. onfrALTVD 28431 is the Virtual Deduction proof from which onfrALT 28061 is derived. The Virtual Deduction proof mirrors the working proof of onfr 4534 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 28431. 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 4481 . 2  |-  (  _E  Fr  On  <->  A. a
( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) ) )
2 simpr 447 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  =/=  (/) )
3 n0 3552 . . . 4  |-  ( a  =/=  (/)  <->  E. x  x  e.  a )
4 onfrALTlem1 28060 . . . . . . 7  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) )
54exp3a 425 . . . . . 6  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
x  e.  a  -> 
( ( a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ) )
6 onfrALTlem2 28058 . . . . . . 7  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  -.  ( a  i^i  x )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) )
76exp3a 425 . . . . . 6  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
x  e.  a  -> 
( -.  ( a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ) )
8 pm2.61 163 . . . . . 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 1367 . . . . 5  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
x  e.  a  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) ) )
109exlimdv 1641 . . . 4  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  ( E. x  x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) )
113, 10syl5bi 208 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
a  =/=  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) )
122, 11mpd 14 . 2  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )
131, 12mpgbir 1555 1  |-  _E  Fr  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1546    = wceq 1647    =/= wne 2529   E.wrex 2629    i^i cin 3237    C_ wss 3238   (/)c0 3543    _E cep 4406    Fr wfr 4452   Oncon0 4495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499
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