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Theorem xpsneng 6943
Description: A set is equinumerous to its cross-product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)
Assertion
Ref Expression
xpsneng  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  { B } )  ~~  A
)

Proof of Theorem xpsneng
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpeq1 4702 . . 3  |-  ( x  =  A  ->  (
x  X.  { y } )  =  ( A  X.  { y } ) )
2 id 19 . . 3  |-  ( x  =  A  ->  x  =  A )
31, 2breq12d 4037 . 2  |-  ( x  =  A  ->  (
( x  X.  {
y } )  ~~  x 
<->  ( A  X.  {
y } )  ~~  A ) )
4 sneq 3652 . . . 4  |-  ( y  =  B  ->  { y }  =  { B } )
54xpeq2d 4712 . . 3  |-  ( y  =  B  ->  ( A  X.  { y } )  =  ( A  X.  { B }
) )
65breq1d 4034 . 2  |-  ( y  =  B  ->  (
( A  X.  {
y } )  ~~  A 
<->  ( A  X.  { B } )  ~~  A
) )
7 vex 2792 . . 3  |-  x  e. 
_V
8 vex 2792 . . 3  |-  y  e. 
_V
97, 8xpsnen 6942 . 2  |-  ( x  X.  { y } )  ~~  x
103, 6, 9vtocl2g 2848 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  { B } )  ~~  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685   {csn 3641   class class class wbr 4024    X. cxp 4686    ~~ cen 6856
This theorem is referenced by:  xp1en  6944  xpsnen2g  6951  xpdom3  6956  disjen  7014  unxpdom2  7067  sucxpdom  7068  uncdadom  7793  cdaun  7794  cdaen  7795  cda1dif  7798  cdacomen  7803  cdaassen  7804  xpcdaen  7805  mapcdaen  7806  cdaxpdom  7811  cdafi  7812  cdainf  7814  infcda1  7815  pwcdadom  7838  isfin4-3  7937  pwcdandom  8285  gchxpidm  8287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-int 3864  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-en 6860
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