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Theorem unixp0 5222
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 3852 . . 3  |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  = 
U. (/) )
2 uni0 3870 . . 3  |-  U. (/)  =  (/)
31, 2syl6eq 2344 . 2  |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  =  (/) )
4 n0 3477 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  <->  E. z  z  e.  ( A  X.  B
) )
5 elxp3 4755 . . . . . 6  |-  ( z  e.  ( A  X.  B )  <->  E. x E. y ( <. x ,  y >.  =  z  /\  <. x ,  y
>.  e.  ( A  X.  B ) ) )
6 elssuni 3871 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  <. x ,  y >.  C_  U. ( A  X.  B ) )
7 vex 2804 . . . . . . . . . 10  |-  x  e. 
_V
8 vex 2804 . . . . . . . . . 10  |-  y  e. 
_V
97, 8opnzi 4259 . . . . . . . . 9  |-  <. x ,  y >.  =/=  (/)
10 ssn0 3500 . . . . . . . . 9  |-  ( (
<. x ,  y >.  C_ 
U. ( A  X.  B )  /\  <. x ,  y >.  =/=  (/) )  ->  U. ( A  X.  B
)  =/=  (/) )
116, 9, 10sylancl 643 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  U. ( A  X.  B )  =/=  (/) )
1211adantl 452 . . . . . . 7  |-  ( (
<. x ,  y >.  =  z  /\  <. x ,  y >.  e.  ( A  X.  B ) )  ->  U. ( A  X.  B )  =/=  (/) )
1312exlimivv 1625 . . . . . 6  |-  ( E. x E. y (
<. x ,  y >.  =  z  /\  <. x ,  y >.  e.  ( A  X.  B ) )  ->  U. ( A  X.  B )  =/=  (/) )
145, 13sylbi 187 . . . . 5  |-  ( z  e.  ( A  X.  B )  ->  U. ( A  X.  B )  =/=  (/) )
1514exlimiv 1624 . . . 4  |-  ( E. z  z  e.  ( A  X.  B )  ->  U. ( A  X.  B )  =/=  (/) )
164, 15sylbi 187 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  U. ( A  X.  B )  =/=  (/) )
1716necon4i 2519 . 2  |-  ( U. ( A  X.  B
)  =  (/)  ->  ( A  X.  B )  =  (/) )
183, 17impbii 180 1  |-  ( ( A  X.  B )  =  (/)  <->  U. ( A  X.  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   <.cop 3656   U.cuni 3843    X. cxp 4703
This theorem is referenced by:  rankxpsuc  7568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-opab 4094  df-xp 4711
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