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Theorem xpmlem 4687
Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)
Assertion
Ref Expression
xpmlem  X.
Distinct variable groups:   ,,,   ,,,

Proof of Theorem xpmlem
StepHypRef Expression
1 eeanv 1804 . . 3
2 vex 2554 . . . . . 6 
_V
3 vex 2554 . . . . . 6 
_V
42, 3opex 3957 . . . . 5  <. ,  >.  _V
5 eleq1 2097 . . . . . 6  <. , 
>.  X. 
<. ,  >.  X.
6 opelxp 4317 . . . . . 6  <. ,  >.  X.
75, 6syl6bb 185 . . . . 5  <. , 
>.  X.
84, 7spcev 2641 . . . 4  X.
98exlimivv 1773 . . 3  X.
101, 9sylbir 125 . 2  X.
11 elxp 4305 . . . . 5  X. 
<. ,  >.
12 simpr 103 . . . . . 6  <. ,  >.
13122eximi 1489 . . . . 5  <. , 
>.
1411, 13sylbi 114 . . . 4  X.
1514exlimiv 1486 . . 3  X.
1615, 1sylib 127 . 2  X.
1710, 16impbii 117 1  X.
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390   <.cop 3370    X. cxp 4286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294
This theorem is referenced by:  xpm  4688
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