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Theorem unixp0 5206
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 3836 . . 3  |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  = 
U. (/) )
2 uni0 3854 . . 3  |-  U. (/)  =  (/)
31, 2syl6eq 2331 . 2  |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  =  (/) )
4 n0 3464 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  <->  E. z  z  e.  ( A  X.  B
) )
5 elxp3 4739 . . . . . 6  |-  ( z  e.  ( A  X.  B )  <->  E. x E. y ( <. x ,  y >.  =  z  /\  <. x ,  y
>.  e.  ( A  X.  B ) ) )
6 elssuni 3855 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  <. x ,  y >.  C_  U. ( A  X.  B ) )
7 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
8 vex 2791 . . . . . . . . . 10  |-  y  e. 
_V
97, 8opnzi 4243 . . . . . . . . 9  |-  <. x ,  y >.  =/=  (/)
10 ssn0 3487 . . . . . . . . 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 1667 . . . . . 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 1666 . . . 4  |-  ( E. z  z  e.  ( A  X.  B )  ->  U. ( A  X.  B )  =/=  (/) )
164, 15sylbi 187 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  U. ( A  X.  B )  =/=  (/) )
1716necon4i 2506 . 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 1528    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   <.cop 3643   U.cuni 3827    X. cxp 4687
This theorem is referenced by:  rankxpsuc  7552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-opab 4078  df-xp 4695
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