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Theorem inawinalem 9015
Description: Lemma for inawina 9016. (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 7499 . . . . 5  |-  ( ~P x  ~<  A  ->  ~P x  ~<_  A )
2 ondomen 8368 . . . . . 6  |-  ( ( A  e.  On  /\  ~P x  ~<_  A )  ->  ~P x  e.  dom  card )
3 isnum2 8276 . . . . . 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 472 . . . 4  |-  ( ( A  e.  On  /\  ~P x  ~<  A )  ->  E. y  e.  On  y  ~~  ~P x )
6 ensdomtr 7609 . . . . . . . . 9  |-  ( ( y  ~~  ~P x  /\  ~P x  ~<  A )  ->  y  ~<  A )
76ad2ant2l 744 . . . . . . . 8  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  -> 
y  ~<  A )
8 sdomel 7620 . . . . . . . . 9  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( y  ~<  A  -> 
y  e.  A ) )
98ad2ant2r 745 . . . . . . . 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 3059 . . . . . . . . . 10  |-  x  e. 
_V
1211canth2 7626 . . . . . . . . 9  |-  x  ~<  ~P x
13 ensym 7520 . . . . . . . . 9  |-  ( y 
~~  ~P x  ->  ~P x  ~~  y )
14 sdomentr 7607 . . . . . . . . 9  |-  ( ( x  ~<  ~P x  /\  ~P x  ~~  y
)  ->  x  ~<  y )
1512, 13, 14sylancr 661 . . . . . . . 8  |-  ( y 
~~  ~P x  ->  x  ~<  y )
1615ad2antlr 725 . . . . . . 7  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  ->  x  ~<  y )
1710, 16jca 530 . . . . . 6  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  -> 
( y  e.  A  /\  x  ~<  y ) )
1817expcom 433 . . . . 5  |-  ( ( A  e.  On  /\  ~P x  ~<  A )  ->  ( ( y  e.  On  /\  y  ~~  ~P x )  -> 
( y  e.  A  /\  x  ~<  y ) ) )
1918reximdv2 2872 . . . 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 432 . 2  |-  ( A  e.  On  ->  ( ~P x  ~<  A  ->  E. y  e.  A  x  ~<  y ) )
2221ralimdv 2811 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 367    e. wcel 1840   A.wral 2751   E.wrex 2752   ~Pcpw 3952   class class class wbr 4392   Oncon0 4819   dom cdm 4940    ~~ cen 7469    ~<_ cdom 7470    ~< csdm 7471   cardccrd 8266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-recs 6997  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-card 8270
This theorem is referenced by:  inawina  9016  tskcard  9107  gruina  9144
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