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Theorem inawinalem 9058
Description: Lemma for inawina 9059. (Contributed by Mario Carneiro, 8-Jun-2014.)
Assertion
Ref Expression
inawinalem  |-  ( A  e.  On  ->  ( A. x  e.  A  ~P x  ~<  A  ->  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
Distinct variable group:    x, A, y

Proof of Theorem inawinalem
StepHypRef Expression
1 sdomdom 7535 . . . . 5  |-  ( ~P x  ~<  A  ->  ~P x  ~<_  A )
2 ondomen 8409 . . . . . 6  |-  ( ( A  e.  On  /\  ~P x  ~<_  A )  ->  ~P x  e.  dom  card )
3 isnum2 8317 . . . . . 6  |-  ( ~P x  e.  dom  card  <->  E. y  e.  On  y  ~~  ~P x )
42, 3sylib 196 . . . . 5  |-  ( ( A  e.  On  /\  ~P x  ~<_  A )  ->  E. y  e.  On  y  ~~  ~P x )
51, 4sylan2 474 . . . 4  |-  ( ( A  e.  On  /\  ~P x  ~<  A )  ->  E. y  e.  On  y  ~~  ~P x )
6 ensdomtr 7645 . . . . . . . . 9  |-  ( ( y  ~~  ~P x  /\  ~P x  ~<  A )  ->  y  ~<  A )
76ad2ant2l 745 . . . . . . . 8  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  -> 
y  ~<  A )
8 sdomel 7656 . . . . . . . . 9  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( y  ~<  A  -> 
y  e.  A ) )
98ad2ant2r 746 . . . . . . . 8  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  -> 
( y  ~<  A  -> 
y  e.  A ) )
107, 9mpd 15 . . . . . . 7  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  -> 
y  e.  A )
11 vex 3111 . . . . . . . . . 10  |-  x  e. 
_V
1211canth2 7662 . . . . . . . . 9  |-  x  ~<  ~P x
13 ensym 7556 . . . . . . . . 9  |-  ( y 
~~  ~P x  ->  ~P x  ~~  y )
14 sdomentr 7643 . . . . . . . . 9  |-  ( ( x  ~<  ~P x  /\  ~P x  ~~  y
)  ->  x  ~<  y )
1512, 13, 14sylancr 663 . . . . . . . 8  |-  ( y 
~~  ~P x  ->  x  ~<  y )
1615ad2antlr 726 . . . . . . 7  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  ->  x  ~<  y )
1710, 16jca 532 . . . . . 6  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  -> 
( y  e.  A  /\  x  ~<  y ) )
1817expcom 435 . . . . 5  |-  ( ( A  e.  On  /\  ~P x  ~<  A )  ->  ( ( y  e.  On  /\  y  ~~  ~P x )  -> 
( y  e.  A  /\  x  ~<  y ) ) )
1918reximdv2 2929 . . . 4  |-  ( ( A  e.  On  /\  ~P x  ~<  A )  ->  ( E. y  e.  On  y  ~~  ~P x  ->  E. y  e.  A  x  ~<  y ) )
205, 19mpd 15 . . 3  |-  ( ( A  e.  On  /\  ~P x  ~<  A )  ->  E. y  e.  A  x  ~<  y )
2120ex 434 . 2  |-  ( A  e.  On  ->  ( ~P x  ~<  A  ->  E. y  e.  A  x  ~<  y ) )
2221ralimdv 2869 1  |-  ( A  e.  On  ->  ( A. x  e.  A  ~P x  ~<  A  ->  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762   A.wral 2809   E.wrex 2810   ~Pcpw 4005   class class class wbr 4442   Oncon0 4873   dom cdm 4994    ~~ cen 7505    ~<_ cdom 7506    ~< csdm 7507   cardccrd 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-recs 7034  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-card 8311
This theorem is referenced by:  inawina  9059  tskcard  9150  gruina  9187
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