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Theorem xpsneng 7594
Description: A set is equinumerous to its Cartesian 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 5008 . . 3  |-  ( x  =  A  ->  (
x  X.  { y } )  =  ( A  X.  { y } ) )
2 id 22 . . 3  |-  ( x  =  A  ->  x  =  A )
31, 2breq12d 4455 . 2  |-  ( x  =  A  ->  (
( x  X.  {
y } )  ~~  x 
<->  ( A  X.  {
y } )  ~~  A ) )
4 sneq 4032 . . . 4  |-  ( y  =  B  ->  { y }  =  { B } )
54xpeq2d 5018 . . 3  |-  ( y  =  B  ->  ( A  X.  { y } )  =  ( A  X.  { B }
) )
65breq1d 4452 . 2  |-  ( y  =  B  ->  (
( A  X.  {
y } )  ~~  A 
<->  ( A  X.  { B } )  ~~  A
) )
7 vex 3111 . . 3  |-  x  e. 
_V
8 vex 3111 . . 3  |-  y  e. 
_V
97, 8xpsnen 7593 . 2  |-  ( x  X.  { y } )  ~~  x
103, 6, 9vtocl2g 3170 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  { B } )  ~~  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   {csn 4022   class class class wbr 4442    X. cxp 4992    ~~ cen 7505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-int 4278  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-en 7509
This theorem is referenced by:  xp1en  7595  xpsnen2g  7602  xpdom3  7607  disjen  7666  unxpdom2  7720  sucxpdom  7721  uncdadom  8542  cdaun  8543  cdaen  8544  cda1dif  8547  cdacomen  8552  cdaassen  8553  xpcdaen  8554  mapcdaen  8555  cdaxpdom  8560  cdafi  8561  cdainf  8563  infcda1  8564  pwcdadom  8587  isfin4-3  8686  pwcdandom  9036  gchxpidm  9038  frlmiscvec  18646
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