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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  |-  ( ( A  X.  B )  =  (/)  <->  U. ( A  X.  B )  =  (/) )

Proof of Theorem unixp0
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unieq 4016 . . 3  |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  = 
U. (/) )
2 uni0 4034 . . 3  |-  U. (/)  =  (/)
31, 2syl6eq 2483 . 2  |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  =  (/) )
4 n0 3629 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  <->  E. z  z  e.  ( A  X.  B
) )
5 elxp3 4920 . . . . . 6  |-  ( z  e.  ( A  X.  B )  <->  E. x E. y ( <. x ,  y >.  =  z  /\  <. x ,  y
>.  e.  ( A  X.  B ) ) )
6 elssuni 4035 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  <. x ,  y >.  C_  U. ( A  X.  B ) )
7 vex 2951 . . . . . . . . . 10  |-  x  e. 
_V
8 vex 2951 . . . . . . . . . 10  |-  y  e. 
_V
97, 8opnzi 4425 . . . . . . . . 9  |-  <. x ,  y >.  =/=  (/)
10 ssn0 3652 . . . . . . . . 9  |-  ( (
<. x ,  y >.  C_ 
U. ( A  X.  B )  /\  <. x ,  y >.  =/=  (/) )  ->  U. ( A  X.  B
)  =/=  (/) )
116, 9, 10sylancl 644 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  U. ( A  X.  B )  =/=  (/) )
1211adantl 453 . . . . . . 7  |-  ( (
<. x ,  y >.  =  z  /\  <. x ,  y >.  e.  ( A  X.  B ) )  ->  U. ( A  X.  B )  =/=  (/) )
1312exlimivv 1645 . . . . . 6  |-  ( E. x E. y (
<. x ,  y >.  =  z  /\  <. x ,  y >.  e.  ( A  X.  B ) )  ->  U. ( A  X.  B )  =/=  (/) )
145, 13sylbi 188 . . . . 5  |-  ( z  e.  ( A  X.  B )  ->  U. ( A  X.  B )  =/=  (/) )
1514exlimiv 1644 . . . 4  |-  ( E. z  z  e.  ( A  X.  B )  ->  U. ( A  X.  B )  =/=  (/) )
164, 15sylbi 188 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  U. ( A  X.  B )  =/=  (/) )
1716necon4i 2658 . 2  |-  ( U. ( A  X.  B
)  =  (/)  ->  ( A  X.  B )  =  (/) )
183, 17impbii 181 1  |-  ( ( A  X.  B )  =  (/)  <->  U. ( A  X.  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   <.cop 3809   U.cuni 4007    X. 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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