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Theorem inawinalem 8311
Description: Lemma for inawina 8312. (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 6889 . . . . 5  |-  ( ~P x  ~<  A  ->  ~P x  ~<_  A )
2 ondomen 7664 . . . . . 6  |-  ( ( A  e.  On  /\  ~P x  ~<_  A )  ->  ~P x  e.  dom  card )
3 isnum2 7578 . . . . . 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 6997 . . . . . . . . 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 7008 . . . . . . . . 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 2791 . . . . . . . . . 10  |-  x  e. 
_V
1211canth2 7014 . . . . . . . . 9  |-  x  ~<  ~P x
13 ensym 6910 . . . . . . . . 9  |-  ( y 
~~  ~P x  ->  ~P x  ~~  y )
14 sdomentr 6995 . . . . . . . . 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 2652 . . . 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 2622 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 1684   A.wral 2543   E.wrex 2544   ~Pcpw 3625   class class class wbr 4023   Oncon0 4392   dom cdm 4689    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568
This theorem is referenced by:  inawina  8312  tskcard  8403  gruina  8440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-card 7572
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