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Theorem onfrALT 27450
Description: The epsilon relation is foundational on the class of ordinal numbers. onfrALT 27450 is an alternate proof of onfr 4403. onfrALTVD 27800 is the Virtual Deduction proof from which onfrALT 27450 is derived. The Virtual Deduction proof mirrors the working proof of onfr 4403 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 27800. 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 4350 . 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 3439 . . . 4  |-  ( a  =/=  (/)  <->  E. x  x  e.  a )
4 onfrALTlem1 27449 . . . . . . 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 27447 . . . . . . 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 2421   E.wrex 2519    i^i cin 3126    C_ wss 3127   (/)c0 3430    _E cep 4275    Fr wfr 4321   Oncon0 4364
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 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368
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