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Theorem xpdom3m 6378
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 6366 . . . . . . 7  |-  ( ( A  e.  V  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~~  A )
213adant2 958 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~~  A )
32ensymd 6330 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~~  ( A  X.  { x }
) )
4 xpexg 4480 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
543adant3 959 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  B
)  e.  _V )
6 simp3 941 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  x  e.  B )
76snssd 3538 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  C_  B )
8 xpss2 4477 . . . . . . 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 6325 . . . . . 6  |-  ( ( A  X.  B )  e.  _V  ->  (
( A  X.  {
x } )  C_  ( A  X.  B
)  ->  ( A  X.  { x } )  ~<_  ( A  X.  B
) ) )
115, 9, 10sylc 61 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~<_  ( A  X.  B ) )
12 endomtr 6337 . . . . 5  |-  ( ( A  ~~  ( A  X.  { x }
)  /\  ( A  X.  { x } )  ~<_  ( A  X.  B
) )  ->  A  ~<_  ( A  X.  B
) )
133, 11, 12syl2anc 403 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~<_  ( A  X.  B ) )
14133expia 1141 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  B  ->  A  ~<_  ( A  X.  B ) ) )
1514exlimdv 1741 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  e.  B  ->  A  ~<_  ( A  X.  B
) ) )
16153impia 1136 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 102    /\ w3a 920   E.wex 1422    e. wcel 1434   _Vcvv 2602    C_ wss 2974   {csn 3406   class class class wbr 3793    X. cxp 4369    ~~ cen 6285    ~<_ cdom 6286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-er 6172  df-en 6288  df-dom 6289
This theorem is referenced by: (None)
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