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Theorem xpdom3m 6848
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 6836 . . . . . . 7  |-  ( ( A  e.  V  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~~  A )
213adant2 1017 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~~  A )
32ensymd 6797 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~~  ( A  X.  { x }
) )
4 xpexg 4752 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
543adant3 1018 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  B
)  e.  _V )
6 simp3 1000 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  x  e.  B )
76snssd 3749 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  C_  B )
8 xpss2 4749 . . . . . . 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 6792 . . . . . 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 6804 . . . . 5  |-  ( ( A  ~~  ( A  X.  { x }
)  /\  ( A  X.  { x } )  ~<_  ( A  X.  B
) )  ->  A  ~<_  ( A  X.  B
) )
133, 11, 12syl2anc 411 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~<_  ( A  X.  B ) )
14133expia 1206 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  B  ->  A  ~<_  ( A  X.  B ) ) )
1514exlimdv 1829 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  e.  B  ->  A  ~<_  ( A  X.  B
) ) )
16153impia 1201 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 104    /\ w3a 979   E.wex 1502    e. wcel 2158   _Vcvv 2749    C_ wss 3141   {csn 3604   class class class wbr 4015    X. cxp 4636    ~~ cen 6752    ~<_ cdom 6753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-er 6549  df-en 6755  df-dom 6756
This theorem is referenced by: (None)
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