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Theorem onfr 4779
Description: The ordinal class is well-founded. This lemma is needed for ordon 6415 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 4726 . 2  |-  (  _E  Fr  On  <->  A. x
( ( x  C_  On  /\  x  =/=  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) ) )
2 n0 3667 . . . 4  |-  ( x  =/=  (/)  <->  E. y  y  e.  x )
3 ineq2 3567 . . . . . . . . . 10  |-  ( z  =  y  ->  (
x  i^i  z )  =  ( x  i^i  y ) )
43eqeq1d 2451 . . . . . . . . 9  |-  ( z  =  y  ->  (
( x  i^i  z
)  =  (/)  <->  ( x  i^i  y )  =  (/) ) )
54rspcev 3094 . . . . . . . 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 3591 . . . . . . . 8  |-  ( x  i^i  y )  C_  x
8 ssel2 3372 . . . . . . . . . . . 12  |-  ( ( x  C_  On  /\  y  e.  x )  ->  y  e.  On )
9 eloni 4750 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  Ord  y )
108, 9syl 16 . . . . . . . . . . 11  |-  ( ( x  C_  On  /\  y  e.  x )  ->  Ord  y )
11 ordfr 4755 . . . . . . . . . . 11  |-  ( Ord  y  ->  _E  Fr  y )
1210, 11syl 16 . . . . . . . . . 10  |-  ( ( x  C_  On  /\  y  e.  x )  ->  _E  Fr  y )
13 inss2 3592 . . . . . . . . . . 11  |-  ( x  i^i  y )  C_  y
14 vex 2996 . . . . . . . . . . . . 13  |-  x  e. 
_V
1514inex1 4454 . . . . . . . . . . . 12  |-  ( x  i^i  y )  e. 
_V
1615epfrc 4727 . . . . . . . . . . 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 1302 . . . . . . . . . 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 3581 . . . . . . . . . . . . 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 3375 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  z  e.  y )
23 ordelss 4756 . . . . . . . . . . . . . . . 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 3576 . . . . . . . . . . . . . . 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 3573 . . . . . . . . . . . . 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 2487 . . . . . . . . . . . 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 2451 . . . . . . . . . . 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 2753 . . . . . . . . . 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 3438 . . . . . . . 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 2713 . . . . . 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 1690 . . . 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 1595 1  |-  _E  Fr  On
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2620   E.wrex 2737    i^i cin 3348    C_ wss 3349   (/)c0 3658    _E cep 4651    Fr wfr 4697   Ord word 4739   Oncon0 4740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-tr 4407  df-eprel 4653  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744
This theorem is referenced by:  ordon  6415
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