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Theorem onfr 4926
Description: The ordinal class is well-founded. This lemma is needed for ordon 6617 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 4873 . 2  |-  (  _E  Fr  On  <->  A. x
( ( x  C_  On  /\  x  =/=  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) ) )
2 n0 3803 . . . 4  |-  ( x  =/=  (/)  <->  E. y  y  e.  x )
3 ineq2 3690 . . . . . . . . . 10  |-  ( z  =  y  ->  (
x  i^i  z )  =  ( x  i^i  y ) )
43eqeq1d 2459 . . . . . . . . 9  |-  ( z  =  y  ->  (
( x  i^i  z
)  =  (/)  <->  ( x  i^i  y )  =  (/) ) )
54rspcev 3210 . . . . . . . 8  |-  ( ( y  e.  x  /\  ( x  i^i  y
)  =  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) )
65adantll 713 . . . . . . 7  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  ( x  i^i  y )  =  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) )
7 inss1 3714 . . . . . . . 8  |-  ( x  i^i  y )  C_  x
8 ssel2 3494 . . . . . . . . . . . 12  |-  ( ( x  C_  On  /\  y  e.  x )  ->  y  e.  On )
9 eloni 4897 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  Ord  y )
108, 9syl 16 . . . . . . . . . . 11  |-  ( ( x  C_  On  /\  y  e.  x )  ->  Ord  y )
11 ordfr 4902 . . . . . . . . . . 11  |-  ( Ord  y  ->  _E  Fr  y )
1210, 11syl 16 . . . . . . . . . 10  |-  ( ( x  C_  On  /\  y  e.  x )  ->  _E  Fr  y )
13 inss2 3715 . . . . . . . . . . 11  |-  ( x  i^i  y )  C_  y
14 vex 3112 . . . . . . . . . . . . 13  |-  x  e. 
_V
1514inex1 4597 . . . . . . . . . . . 12  |-  ( x  i^i  y )  e. 
_V
1615epfrc 4874 . . . . . . . . . . 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 1312 . . . . . . . . . 10  |-  ( (  _E  Fr  y  /\  ( x  i^i  y
)  =/=  (/) )  ->  E. z  e.  (
x  i^i  y )
( ( x  i^i  y )  i^i  z
)  =  (/) )
1812, 17sylan 471 . . . . . . . . 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 3704 . . . . . . . . . . . . 13  |-  ( ( x  i^i  y )  i^i  z )  =  ( x  i^i  (
y  i^i  z )
)
2010adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  Ord  y )
21 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  z  e.  ( x  i^i  y
) )
2213, 21sseldi 3497 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  z  e.  y )
23 ordelss 4903 . . . . . . . . . . . . . . . 16  |-  ( ( Ord  y  /\  z  e.  y )  ->  z  C_  y )
2420, 22, 23syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  z  C_  y )
25 dfss1 3699 . . . . . . . . . . . . . . 15  |-  ( z 
C_  y  <->  ( y  i^i  z )  =  z )
2624, 25sylib 196 . . . . . . . . . . . . . 14  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  ( y  i^i  z )  =  z )
2726ineq2d 3696 . . . . . . . . . . . . 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 2510 . . . . . . . . . . . 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 2459 . . . . . . . . . . 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 2965 . . . . . . . . . 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 465 . . . . . . . . 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 210 . . . . . . . 8  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  ( x  i^i  y )  =/=  (/) )  ->  E. z  e.  (
x  i^i  y )
( x  i^i  z
)  =  (/) )
33 ssrexv 3561 . . . . . . . 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 63 . . . . . . 7  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  ( x  i^i  y )  =/=  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) )
356, 34pm2.61dane 2775 . . . . . 6  |-  ( ( x  C_  On  /\  y  e.  x )  ->  E. z  e.  x  ( x  i^i  z )  =  (/) )
3635ex 434 . . . . 5  |-  ( x 
C_  On  ->  ( y  e.  x  ->  E. z  e.  x  ( x  i^i  z )  =  (/) ) )
3736exlimdv 1725 . . . 4  |-  ( x 
C_  On  ->  ( E. y  y  e.  x  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) ) )
382, 37syl5bi 217 . . 3  |-  ( x 
C_  On  ->  ( x  =/=  (/)  ->  E. z  e.  x  ( x  i^i  z )  =  (/) ) )
3938imp 429 . 2  |-  ( ( x  C_  On  /\  x  =/=  (/) )  ->  E. z  e.  x  ( x  i^i  z )  =  (/) )
401, 39mpgbir 1623 1  |-  _E  Fr  On
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   E.wrex 2808    i^i cin 3470    C_ wss 3471   (/)c0 3793    _E cep 4798    Fr wfr 4844   Ord word 4886   Oncon0 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891
This theorem is referenced by:  ordon  6617
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