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Theorem inawinalem 8327
Description: Lemma for inawina 8328. (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 6905 . . . . 5  |-  ( ~P x  ~<  A  ->  ~P x  ~<_  A )
2 ondomen 7680 . . . . . 6  |-  ( ( A  e.  On  /\  ~P x  ~<_  A )  ->  ~P x  e.  dom  card )
3 isnum2 7594 . . . . . 6  |-  ( ~P x  e.  dom  card  <->  E. y  e.  On  y  ~~  ~P x )
42, 3sylib 188 . . . . 5  |-  ( ( A  e.  On  /\  ~P x  ~<_  A )  ->  E. y  e.  On  y  ~~  ~P x )
51, 4sylan2 460 . . . 4  |-  ( ( A  e.  On  /\  ~P x  ~<  A )  ->  E. y  e.  On  y  ~~  ~P x )
6 ensdomtr 7013 . . . . . . . . 9  |-  ( ( y  ~~  ~P x  /\  ~P x  ~<  A )  ->  y  ~<  A )
76ad2ant2l 726 . . . . . . . 8  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  -> 
y  ~<  A )
8 sdomel 7024 . . . . . . . . 9  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( y  ~<  A  -> 
y  e.  A ) )
98ad2ant2r 727 . . . . . . . 8  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  -> 
( y  ~<  A  -> 
y  e.  A ) )
107, 9mpd 14 . . . . . . 7  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  -> 
y  e.  A )
11 vex 2804 . . . . . . . . . 10  |-  x  e. 
_V
1211canth2 7030 . . . . . . . . 9  |-  x  ~<  ~P x
13 ensym 6926 . . . . . . . . 9  |-  ( y 
~~  ~P x  ->  ~P x  ~~  y )
14 sdomentr 7011 . . . . . . . . 9  |-  ( ( x  ~<  ~P x  /\  ~P x  ~~  y
)  ->  x  ~<  y )
1512, 13, 14sylancr 644 . . . . . . . 8  |-  ( y 
~~  ~P x  ->  x  ~<  y )
1615ad2antlr 707 . . . . . . 7  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  ->  x  ~<  y )
1710, 16jca 518 . . . . . 6  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  -> 
( y  e.  A  /\  x  ~<  y ) )
1817expcom 424 . . . . 5  |-  ( ( A  e.  On  /\  ~P x  ~<  A )  ->  ( ( y  e.  On  /\  y  ~~  ~P x )  -> 
( y  e.  A  /\  x  ~<  y ) ) )
1918reximdv2 2665 . . . 4  |-  ( ( A  e.  On  /\  ~P x  ~<  A )  ->  ( E. y  e.  On  y  ~~  ~P x  ->  E. y  e.  A  x  ~<  y ) )
205, 19mpd 14 . . 3  |-  ( ( A  e.  On  /\  ~P x  ~<  A )  ->  E. y  e.  A  x  ~<  y )
2120ex 423 . 2  |-  ( A  e.  On  ->  ( ~P x  ~<  A  ->  E. y  e.  A  x  ~<  y ) )
2221ralimdv 2635 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 set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   A.wral 2556   E.wrex 2557   ~Pcpw 3638   class class class wbr 4039   Oncon0 4408   dom cdm 4705    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   cardccrd 7584
This theorem is referenced by:  inawina  8328  tskcard  8419  gruina  8456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-card 7588
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