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Theorem xpsneng 7622
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

Proof of Theorem xpsneng
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpeq1 5018 . . 3
2 id 22 . . 3
31, 2breq12d 4465 . 2
4 sneq 4039 . . . 4
54xpeq2d 5028 . . 3
65breq1d 4462 . 2
7 vex 3112 . . 3
8 vex 3112 . . 3
97, 8xpsnen 7621 . 2
103, 6, 9vtocl2g 3171 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  {csn 4029   class class class wbr 4452  X.cxp 5002   cen 7533
This theorem is referenced by:  xp1en  7623  xpsnen2g  7630  xpdom3  7635  disjen  7694  unxpdom2  7748  sucxpdom  7749  uncdadom  8572  cdaun  8573  cdaen  8574  cda1dif  8577  cdacomen  8582  cdaassen  8583  xpcdaen  8584  mapcdaen  8585  cdaxpdom  8590  cdafi  8591  cdainf  8593  infcda1  8594  pwcdadom  8617  isfin4-3  8716  pwcdandom  9066  gchxpidm  9068  frlmiscvec  18884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-int 4287  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-en 7537
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