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Theorem xpcomen 6949
 Description: Commutative law for equinumerosity of cross product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
xpcomen.1
xpcomen.2
Assertion
Ref Expression
xpcomen
Dummy variable is distinct from all other variables.

Proof of Theorem xpcomen
StepHypRef Expression
1 xpcomen.1 . . 3
2 xpcomen.2 . . 3
31, 2xpex 4801 . 2
42, 1xpex 4801 . 2
5 eqid 2285 . . 3
65xpcomf1o 6947 . 2
7 f1oen2g 6874 . 2
83, 4, 6, 7mp3an 1279 1
 Colors of variables: wff set class Syntax hints:   wcel 1685  cvv 2790  csn 3642  cuni 3829   class class class wbr 4025   cmpt 4079   cxp 4687  ccnv 4688  wf1o 5221   cen 6856 This theorem is referenced by:  xpcomeng  6950  hashxplem  11380 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-1st 6084  df-2nd 6085  df-en 6860
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