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Theorem xpsneng 7185
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 4884 . . 3  |-  ( x  =  A  ->  (
x  X.  { y } )  =  ( A  X.  { y } ) )
2 id 20 . . 3  |-  ( x  =  A  ->  x  =  A )
31, 2breq12d 4217 . 2  |-  ( x  =  A  ->  (
( x  X.  {
y } )  ~~  x 
<->  ( A  X.  {
y } )  ~~  A ) )
4 sneq 3817 . . . 4  |-  ( y  =  B  ->  { y }  =  { B } )
54xpeq2d 4894 . . 3  |-  ( y  =  B  ->  ( A  X.  { y } )  =  ( A  X.  { B }
) )
65breq1d 4214 . 2  |-  ( y  =  B  ->  (
( A  X.  {
y } )  ~~  A 
<->  ( A  X.  { B } )  ~~  A
) )
7 vex 2951 . . 3  |-  x  e. 
_V
8 vex 2951 . . 3  |-  y  e. 
_V
97, 8xpsnen 7184 . 2  |-  ( x  X.  { y } )  ~~  x
103, 6, 9vtocl2g 3007 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  { B } )  ~~  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3806   class class class wbr 4204    X. cxp 4868    ~~ cen 7098
This theorem is referenced by:  xp1en  7186  xpsnen2g  7193  xpdom3  7198  disjen  7256  unxpdom2  7309  sucxpdom  7310  uncdadom  8043  cdaun  8044  cdaen  8045  cda1dif  8048  cdacomen  8053  cdaassen  8054  xpcdaen  8055  mapcdaen  8056  cdaxpdom  8061  cdafi  8062  cdainf  8064  infcda1  8065  pwcdadom  8088  isfin4-3  8187  pwcdandom  8534  gchxpidm  8536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-en 7102
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