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Theorem unixp0 5345
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 3968 . . 3  |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  = 
U. (/) )
2 uni0 3986 . . 3  |-  U. (/)  =  (/)
31, 2syl6eq 2437 . 2  |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  =  (/) )
4 n0 3582 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  <->  E. z  z  e.  ( A  X.  B
) )
5 elxp3 4870 . . . . . 6  |-  ( z  e.  ( A  X.  B )  <->  E. x E. y ( <. x ,  y >.  =  z  /\  <. x ,  y
>.  e.  ( A  X.  B ) ) )
6 elssuni 3987 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  <. x ,  y >.  C_  U. ( A  X.  B ) )
7 vex 2904 . . . . . . . . . 10  |-  x  e. 
_V
8 vex 2904 . . . . . . . . . 10  |-  y  e. 
_V
97, 8opnzi 4376 . . . . . . . . 9  |-  <. x ,  y >.  =/=  (/)
10 ssn0 3605 . . . . . . . . 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 1642 . . . . . 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 1641 . . . 4  |-  ( E. z  z  e.  ( A  X.  B )  ->  U. ( A  X.  B )  =/=  (/) )
164, 15sylbi 188 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  U. ( A  X.  B )  =/=  (/) )
1716necon4i 2612 . 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 1547    = wceq 1649    e. wcel 1717    =/= wne 2552    C_ wss 3265   (/)c0 3573   <.cop 3762   U.cuni 3959    X. cxp 4818
This theorem is referenced by:  rankxpsuc  7741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-opab 4210  df-xp 4826
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