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Theorem unixp0im 5145
Description: The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)
Assertion
Ref Expression
unixp0im  |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  =  (/) )

Proof of Theorem unixp0im
StepHypRef Expression
1 unieq 3803 . 2  |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  = 
U. (/) )
2 uni0 3821 . 2  |-  U. (/)  =  (/)
31, 2eqtrdi 2219 1  |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348   (/)c0 3414   U.cuni 3794    X. cxp 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3587  df-uni 3795
This theorem is referenced by: (None)
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