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Theorem onfr 4562
Description: The ordinal class is well-founded. This lemma is needed for ordon 4704 in order to eliminate the need for the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
Assertion
Ref Expression
onfr  |-  _E  Fr  On

Proof of Theorem onfr
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfepfr 4509 . 2  |-  (  _E  Fr  On  <->  A. x
( ( x  C_  On  /\  x  =/=  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) ) )
2 n0 3581 . . . 4  |-  ( x  =/=  (/)  <->  E. y  y  e.  x )
3 ineq2 3480 . . . . . . . . . 10  |-  ( z  =  y  ->  (
x  i^i  z )  =  ( x  i^i  y ) )
43eqeq1d 2396 . . . . . . . . 9  |-  ( z  =  y  ->  (
( x  i^i  z
)  =  (/)  <->  ( x  i^i  y )  =  (/) ) )
54rspcev 2996 . . . . . . . 8  |-  ( ( y  e.  x  /\  ( x  i^i  y
)  =  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) )
65adantll 695 . . . . . . 7  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  ( x  i^i  y )  =  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) )
7 inss1 3505 . . . . . . . 8  |-  ( x  i^i  y )  C_  x
8 ssel2 3287 . . . . . . . . . . . 12  |-  ( ( x  C_  On  /\  y  e.  x )  ->  y  e.  On )
9 eloni 4533 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  Ord  y )
108, 9syl 16 . . . . . . . . . . 11  |-  ( ( x  C_  On  /\  y  e.  x )  ->  Ord  y )
11 ordfr 4538 . . . . . . . . . . 11  |-  ( Ord  y  ->  _E  Fr  y )
1210, 11syl 16 . . . . . . . . . 10  |-  ( ( x  C_  On  /\  y  e.  x )  ->  _E  Fr  y )
13 inss2 3506 . . . . . . . . . . 11  |-  ( x  i^i  y )  C_  y
14 vex 2903 . . . . . . . . . . . . 13  |-  x  e. 
_V
1514inex1 4286 . . . . . . . . . . . 12  |-  ( x  i^i  y )  e. 
_V
1615epfrc 4510 . . . . . . . . . . 11  |-  ( (  _E  Fr  y  /\  ( x  i^i  y
)  C_  y  /\  ( x  i^i  y
)  =/=  (/) )  ->  E. z  e.  (
x  i^i  y )
( ( x  i^i  y )  i^i  z
)  =  (/) )
1713, 16mp3an2 1267 . . . . . . . . . 10  |-  ( (  _E  Fr  y  /\  ( x  i^i  y
)  =/=  (/) )  ->  E. z  e.  (
x  i^i  y )
( ( x  i^i  y )  i^i  z
)  =  (/) )
1812, 17sylan 458 . . . . . . . . 9  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  ( x  i^i  y )  =/=  (/) )  ->  E. z  e.  (
x  i^i  y )
( ( x  i^i  y )  i^i  z
)  =  (/) )
19 inass 3495 . . . . . . . . . . . . 13  |-  ( ( x  i^i  y )  i^i  z )  =  ( x  i^i  (
y  i^i  z )
)
2010adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  Ord  y )
21 simpr 448 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  z  e.  ( x  i^i  y
) )
2213, 21sseldi 3290 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  z  e.  y )
23 ordelss 4539 . . . . . . . . . . . . . . . 16  |-  ( ( Ord  y  /\  z  e.  y )  ->  z  C_  y )
2420, 22, 23syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  z  C_  y )
25 dfss1 3489 . . . . . . . . . . . . . . 15  |-  ( z 
C_  y  <->  ( y  i^i  z )  =  z )
2624, 25sylib 189 . . . . . . . . . . . . . 14  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  ( y  i^i  z )  =  z )
2726ineq2d 3486 . . . . . . . . . . . . 13  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  ( x  i^i  ( y  i^i  z
) )  =  ( x  i^i  z ) )
2819, 27syl5eq 2432 . . . . . . . . . . . 12  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  ( (
x  i^i  y )  i^i  z )  =  ( x  i^i  z ) )
2928eqeq1d 2396 . . . . . . . . . . 11  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  ( (
( x  i^i  y
)  i^i  z )  =  (/)  <->  ( x  i^i  z )  =  (/) ) )
3029rexbidva 2667 . . . . . . . . . 10  |-  ( ( x  C_  On  /\  y  e.  x )  ->  ( E. z  e.  (
x  i^i  y )
( ( x  i^i  y )  i^i  z
)  =  (/)  <->  E. z  e.  ( x  i^i  y
) ( x  i^i  z )  =  (/) ) )
3130adantr 452 . . . . . . . . 9  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  ( x  i^i  y )  =/=  (/) )  -> 
( E. z  e.  ( x  i^i  y
) ( ( x  i^i  y )  i^i  z )  =  (/)  <->  E. z  e.  ( x  i^i  y ) ( x  i^i  z )  =  (/) ) )
3218, 31mpbid 202 . . . . . . . 8  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  ( x  i^i  y )  =/=  (/) )  ->  E. z  e.  (
x  i^i  y )
( x  i^i  z
)  =  (/) )
33 ssrexv 3352 . . . . . . . 8  |-  ( ( x  i^i  y ) 
C_  x  ->  ( E. z  e.  (
x  i^i  y )
( x  i^i  z
)  =  (/)  ->  E. z  e.  x  ( x  i^i  z )  =  (/) ) )
347, 32, 33mpsyl 61 . . . . . . 7  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  ( x  i^i  y )  =/=  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) )
356, 34pm2.61dane 2629 . . . . . 6  |-  ( ( x  C_  On  /\  y  e.  x )  ->  E. z  e.  x  ( x  i^i  z )  =  (/) )
3635ex 424 . . . . 5  |-  ( x 
C_  On  ->  ( y  e.  x  ->  E. z  e.  x  ( x  i^i  z )  =  (/) ) )
3736exlimdv 1643 . . . 4  |-  ( x 
C_  On  ->  ( E. y  y  e.  x  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) ) )
382, 37syl5bi 209 . . 3  |-  ( x 
C_  On  ->  ( x  =/=  (/)  ->  E. z  e.  x  ( x  i^i  z )  =  (/) ) )
3938imp 419 . 2  |-  ( ( x  C_  On  /\  x  =/=  (/) )  ->  E. z  e.  x  ( x  i^i  z )  =  (/) )
401, 39mpgbir 1556 1  |-  _E  Fr  On
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651    i^i cin 3263    C_ wss 3264   (/)c0 3572    _E cep 4434    Fr wfr 4480   Ord word 4522   Oncon0 4523
This theorem is referenced by:  ordon  4704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527
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