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Theorem xpdom3m 6696
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 6684 . . . . . . 7  |-  ( ( A  e.  V  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~~  A )
213adant2 985 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~~  A )
32ensymd 6645 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~~  ( A  X.  { x }
) )
4 xpexg 4623 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
543adant3 986 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  B
)  e.  _V )
6 simp3 968 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  x  e.  B )
76snssd 3635 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  C_  B )
8 xpss2 4620 . . . . . . 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 6640 . . . . . 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 6652 . . . . 5  |-  ( ( A  ~~  ( A  X.  { x }
)  /\  ( A  X.  { x } )  ~<_  ( A  X.  B
) )  ->  A  ~<_  ( A  X.  B
) )
133, 11, 12syl2anc 408 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~<_  ( A  X.  B ) )
14133expia 1168 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  B  ->  A  ~<_  ( A  X.  B ) ) )
1514exlimdv 1775 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  e.  B  ->  A  ~<_  ( A  X.  B
) ) )
16153impia 1163 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 947   E.wex 1453    e. wcel 1465   _Vcvv 2660    C_ wss 3041   {csn 3497   class class class wbr 3899    X. cxp 4507    ~~ cen 6600    ~<_ cdom 6601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-er 6397  df-en 6603  df-dom 6604
This theorem is referenced by: (None)
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