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Theorem onfr 4468
Description: The ordinal class is well-founded. This lemma is needed for ordon 4611 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 4415 . 2  |-  (  _E  Fr  On  <->  A. x
( ( x  C_  On  /\  x  =/=  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) ) )
2 n0 3498 . . . 4  |-  ( x  =/=  (/)  <->  E. y  y  e.  x )
3 ineq2 3398 . . . . . . . . . 10  |-  ( z  =  y  ->  (
x  i^i  z )  =  ( x  i^i  y ) )
43eqeq1d 2324 . . . . . . . . 9  |-  ( z  =  y  ->  (
( x  i^i  z
)  =  (/)  <->  ( x  i^i  y )  =  (/) ) )
54rspcev 2918 . . . . . . . 8  |-  ( ( y  e.  x  /\  ( x  i^i  y
)  =  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) )
65adantll 694 . . . . . . 7  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  ( x  i^i  y )  =  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) )
7 ssel2 3209 . . . . . . . . . . . 12  |-  ( ( x  C_  On  /\  y  e.  x )  ->  y  e.  On )
8 eloni 4439 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  Ord  y )
97, 8syl 15 . . . . . . . . . . 11  |-  ( ( x  C_  On  /\  y  e.  x )  ->  Ord  y )
10 ordfr 4444 . . . . . . . . . . 11  |-  ( Ord  y  ->  _E  Fr  y )
119, 10syl 15 . . . . . . . . . 10  |-  ( ( x  C_  On  /\  y  e.  x )  ->  _E  Fr  y )
12 inss2 3424 . . . . . . . . . . 11  |-  ( x  i^i  y )  C_  y
13 vex 2825 . . . . . . . . . . . . 13  |-  x  e. 
_V
1413inex1 4192 . . . . . . . . . . . 12  |-  ( x  i^i  y )  e. 
_V
1514epfrc 4416 . . . . . . . . . . 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
)  =  (/) )
1612, 15mp3an2 1265 . . . . . . . . . 10  |-  ( (  _E  Fr  y  /\  ( x  i^i  y
)  =/=  (/) )  ->  E. z  e.  (
x  i^i  y )
( ( x  i^i  y )  i^i  z
)  =  (/) )
1711, 16sylan 457 . . . . . . . . 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
)  =  (/) )
18 inass 3413 . . . . . . . . . . . . 13  |-  ( ( x  i^i  y )  i^i  z )  =  ( x  i^i  (
y  i^i  z )
)
199adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  Ord  y )
20 simpr 447 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  z  e.  ( x  i^i  y
) )
2112, 20sseldi 3212 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  z  e.  y )
22 ordelss 4445 . . . . . . . . . . . . . . . 16  |-  ( ( Ord  y  /\  z  e.  y )  ->  z  C_  y )
2319, 21, 22syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  z  C_  y )
24 dfss1 3407 . . . . . . . . . . . . . . 15  |-  ( z 
C_  y  <->  ( y  i^i  z )  =  z )
2523, 24sylib 188 . . . . . . . . . . . . . 14  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  ( y  i^i  z )  =  z )
2625ineq2d 3404 . . . . . . . . . . . . 13  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  ( x  i^i  ( y  i^i  z
) )  =  ( x  i^i  z ) )
2718, 26syl5eq 2360 . . . . . . . . . . . 12  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  ( (
x  i^i  y )  i^i  z )  =  ( x  i^i  z ) )
2827eqeq1d 2324 . . . . . . . . . . 11  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  z  e.  ( x  i^i  y ) )  ->  ( (
( x  i^i  y
)  i^i  z )  =  (/)  <->  ( x  i^i  z )  =  (/) ) )
2928rexbidva 2594 . . . . . . . . . 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 )  =  (/) ) )
3029adantr 451 . . . . . . . . 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 )  =  (/) ) )
3117, 30mpbid 201 . . . . . . . 8  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  ( x  i^i  y )  =/=  (/) )  ->  E. z  e.  (
x  i^i  y )
( x  i^i  z
)  =  (/) )
32 inss1 3423 . . . . . . . . 9  |-  ( x  i^i  y )  C_  x
33 ssrexv 3272 . . . . . . . . 9  |-  ( ( 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 )  =  (/) ) )
3432, 33ax-mp 8 . . . . . . . 8  |-  ( E. z  e.  ( x  i^i  y ) ( x  i^i  z )  =  (/)  ->  E. z  e.  x  ( x  i^i  z )  =  (/) )
3531, 34syl 15 . . . . . . 7  |-  ( ( ( x  C_  On  /\  y  e.  x )  /\  ( x  i^i  y )  =/=  (/) )  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) )
366, 35pm2.61dane 2557 . . . . . 6  |-  ( ( x  C_  On  /\  y  e.  x )  ->  E. z  e.  x  ( x  i^i  z )  =  (/) )
3736ex 423 . . . . 5  |-  ( x 
C_  On  ->  ( y  e.  x  ->  E. z  e.  x  ( x  i^i  z )  =  (/) ) )
3837exlimdv 1627 . . . 4  |-  ( x 
C_  On  ->  ( E. y  y  e.  x  ->  E. z  e.  x  ( x  i^i  z
)  =  (/) ) )
392, 38syl5bi 208 . . 3  |-  ( x 
C_  On  ->  ( x  =/=  (/)  ->  E. z  e.  x  ( x  i^i  z )  =  (/) ) )
4039imp 418 . 2  |-  ( ( x  C_  On  /\  x  =/=  (/) )  ->  E. z  e.  x  ( x  i^i  z )  =  (/) )
411, 40mpgbir 1541 1  |-  _E  Fr  On
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1532    = wceq 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578    i^i cin 3185    C_ wss 3186   (/)c0 3489    _E cep 4340    Fr wfr 4386   Ord word 4428   Oncon0 4429
This theorem is referenced by:  ordon  4611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433
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