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Theorem xpdom3m 6679
Description: A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.)
Assertion
Ref Expression
xpdom3m  |-  ( ( A  e.  V  /\  B  e.  W  /\  E. x  x  e.  B
)  ->  A  ~<_  ( A  X.  B ) )
Distinct variable groups:    x, A    x, B    x, V    x, W

Proof of Theorem xpdom3m
StepHypRef Expression
1 xpsneng 6667 . . . . . . 7  |-  ( ( A  e.  V  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~~  A )
213adant2 981 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~~  A )
32ensymd 6629 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~~  ( A  X.  { x }
) )
4 xpexg 4611 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
543adant3 982 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  B
)  e.  _V )
6 simp3 964 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  x  e.  B )
76snssd 3629 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  C_  B )
8 xpss2 4608 . . . . . . 7  |-  ( { x }  C_  B  ->  ( A  X.  {
x } )  C_  ( A  X.  B
) )
97, 8syl 14 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  {
x } )  C_  ( A  X.  B
) )
10 ssdomg 6624 . . . . . 6  |-  ( ( A  X.  B )  e.  _V  ->  (
( A  X.  {
x } )  C_  ( A  X.  B
)  ->  ( A  X.  { x } )  ~<_  ( A  X.  B
) ) )
115, 9, 10sylc 62 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~<_  ( A  X.  B ) )
12 endomtr 6636 . . . . 5  |-  ( ( A  ~~  ( A  X.  { x }
)  /\  ( A  X.  { x } )  ~<_  ( A  X.  B
) )  ->  A  ~<_  ( A  X.  B
) )
133, 11, 12syl2anc 406 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~<_  ( A  X.  B ) )
14133expia 1164 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  B  ->  A  ~<_  ( A  X.  B ) ) )
1514exlimdv 1771 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  e.  B  ->  A  ~<_  ( A  X.  B
) ) )
16153impia 1159 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  E. x  x  e.  B
)  ->  A  ~<_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 943   E.wex 1449    e. wcel 1461   _Vcvv 2655    C_ wss 3035   {csn 3491   class class class wbr 3893    X. cxp 4495    ~~ cen 6584    ~<_ cdom 6585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-er 6381  df-en 6587  df-dom 6588
This theorem is referenced by: (None)
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