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Theorem xpexr2m 4980
 Description: If a nonempty cross product is a set, so are both of its components. (Contributed by Jim Kingdon, 14-Dec-2018.)
Assertion
Ref Expression
xpexr2m
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem xpexr2m
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpm 4960 . 2
2 dmxpm 4759 . . . . . 6
32adantl 275 . . . . 5
4 dmexg 4803 . . . . . 6
54adantr 274 . . . . 5
63, 5eqeltrrd 2217 . . . 4
7 rnxpm 4968 . . . . . 6
87adantl 275 . . . . 5
9 rnexg 4804 . . . . . 6
109adantr 274 . . . . 5
118, 10eqeltrrd 2217 . . . 4
126, 11anim12dan 589 . . 3
1312ancom2s 555 . 2
141, 13sylan2br 286 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1331  wex 1468   wcel 1480  cvv 2686   cxp 4537   cdm 4539   crn 4540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550 This theorem is referenced by: (None)
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