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Theorem xpdom3 7655
Description: A set is dominated by its Cartesian product with a nonempty set. Exercise 6 of [Suppes] p. 98. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
xpdom3  |-  ( ( A  e.  V  /\  B  e.  W  /\  B  =/=  (/) )  ->  A  ~<_  ( A  X.  B
) )

Proof of Theorem xpdom3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 n0 3750 . . 3  |-  ( B  =/=  (/)  <->  E. x  x  e.  B )
2 xpsneng 7642 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~~  A )
323adant2 1018 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~~  A )
43ensymd 7606 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~~  ( A  X.  { x }
) )
5 xpexg 6586 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
653adant3 1019 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  B
)  e.  _V )
7 simp3 1001 . . . . . . . . 9  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  x  e.  B )
87snssd 4119 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  C_  B )
9 xpss2 4935 . . . . . . . 8  |-  ( { x }  C_  B  ->  ( A  X.  {
x } )  C_  ( A  X.  B
) )
108, 9syl 17 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  {
x } )  C_  ( A  X.  B
) )
11 ssdomg 7601 . . . . . . 7  |-  ( ( A  X.  B )  e.  _V  ->  (
( A  X.  {
x } )  C_  ( A  X.  B
)  ->  ( A  X.  { x } )  ~<_  ( A  X.  B
) ) )
126, 10, 11sylc 61 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  X.  {
x } )  ~<_  ( A  X.  B ) )
13 endomtr 7613 . . . . . 6  |-  ( ( A  ~~  ( A  X.  { x }
)  /\  ( A  X.  { x } )  ~<_  ( A  X.  B
) )  ->  A  ~<_  ( A  X.  B
) )
144, 12, 13syl2anc 661 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~<_  ( A  X.  B ) )
15143expia 1201 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  B  ->  A  ~<_  ( A  X.  B ) ) )
1615exlimdv 1747 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  e.  B  ->  A  ~<_  ( A  X.  B
) ) )
171, 16syl5bi 219 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =/=  (/)  ->  A  ~<_  ( A  X.  B
) ) )
18173impia 1196 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  B  =/=  (/) )  ->  A  ~<_  ( A  X.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 976   E.wex 1635    e. wcel 1844    =/= wne 2600   _Vcvv 3061    C_ wss 3416   (/)c0 3740   {csn 3974   class class class wbr 4397    X. cxp 4823    ~~ cen 7553    ~<_ cdom 7554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-int 4230  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-er 7350  df-en 7557  df-dom 7558
This theorem is referenced by:  mapdom2  7728  xpfir  7779  infxpabs  8626
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