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Theorem unixp0 5395
 Description: A cross product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)
Assertion
Ref Expression
unixp0

Proof of Theorem unixp0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unieq 4016 . . 3
2 uni0 4034 . . 3
31, 2syl6eq 2483 . 2
4 n0 3629 . . . 4
5 elxp3 4920 . . . . . 6
6 elssuni 4035 . . . . . . . . 9
7 vex 2951 . . . . . . . . . 10
8 vex 2951 . . . . . . . . . 10
97, 8opnzi 4425 . . . . . . . . 9
10 ssn0 3652 . . . . . . . . 9
116, 9, 10sylancl 644 . . . . . . . 8
1211adantl 453 . . . . . . 7
1312exlimivv 1645 . . . . . 6
145, 13sylbi 188 . . . . 5
1514exlimiv 1644 . . . 4
164, 15sylbi 188 . . 3
1716necon4i 2658 . 2
183, 17impbii 181 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359  wex 1550   wceq 1652   wcel 1725   wne 2598   wss 3312  c0 3620  cop 3809  cuni 4007   cxp 4868 This theorem is referenced by:  rankxpsuc  7798 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 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-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-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-opab 4259  df-xp 4876
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