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Theorem inawinalem 9132
Description: Lemma for inawina 9133. (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 7615 . . . . 5  |-  ( ~P x  ~<  A  ->  ~P x  ~<_  A )
2 ondomen 8486 . . . . . 6  |-  ( ( A  e.  On  /\  ~P x  ~<_  A )  ->  ~P x  e.  dom  card )
3 isnum2 8397 . . . . . 6  |-  ( ~P x  e.  dom  card  <->  E. y  e.  On  y  ~~  ~P x )
42, 3sylib 201 . . . . 5  |-  ( ( A  e.  On  /\  ~P x  ~<_  A )  ->  E. y  e.  On  y  ~~  ~P x )
51, 4sylan2 482 . . . 4  |-  ( ( A  e.  On  /\  ~P x  ~<  A )  ->  E. y  e.  On  y  ~~  ~P x )
6 ensdomtr 7726 . . . . . . . . 9  |-  ( ( y  ~~  ~P x  /\  ~P x  ~<  A )  ->  y  ~<  A )
76ad2ant2l 760 . . . . . . . 8  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  -> 
y  ~<  A )
8 sdomel 7737 . . . . . . . . 9  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( y  ~<  A  -> 
y  e.  A ) )
98ad2ant2r 761 . . . . . . . 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 3034 . . . . . . . . . 10  |-  x  e. 
_V
1211canth2 7743 . . . . . . . . 9  |-  x  ~<  ~P x
13 ensym 7636 . . . . . . . . 9  |-  ( y 
~~  ~P x  ->  ~P x  ~~  y )
14 sdomentr 7724 . . . . . . . . 9  |-  ( ( x  ~<  ~P x  /\  ~P x  ~~  y
)  ->  x  ~<  y )
1512, 13, 14sylancr 676 . . . . . . . 8  |-  ( y 
~~  ~P x  ->  x  ~<  y )
1615ad2antlr 741 . . . . . . 7  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  ->  x  ~<  y )
1710, 16jca 541 . . . . . 6  |-  ( ( ( y  e.  On  /\  y  ~~  ~P x
)  /\  ( A  e.  On  /\  ~P x  ~<  A ) )  -> 
( y  e.  A  /\  x  ~<  y ) )
1817expcom 442 . . . . 5  |-  ( ( A  e.  On  /\  ~P x  ~<  A )  ->  ( ( y  e.  On  /\  y  ~~  ~P x )  -> 
( y  e.  A  /\  x  ~<  y ) ) )
1918reximdv2 2855 . . . 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 441 . 2  |-  ( A  e.  On  ->  ( ~P x  ~<  A  ->  E. y  e.  A  x  ~<  y ) )
2221ralimdv 2806 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 376    e. wcel 1904   A.wral 2756   E.wrex 2757   ~Pcpw 3942   class class class wbr 4395   dom cdm 4839   Oncon0 5430    ~~ cen 7584    ~<_ cdom 7585    ~< csdm 7586   cardccrd 8387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-wrecs 7046  df-recs 7108  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-card 8391
This theorem is referenced by:  inawina  9133  tskcard  9224  gruina  9261
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