Theorem unixp0im 5083

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

Step	Hyp	Ref	Expression
1		unieq 3753	. 2 |- ( ( A X. B ) = (/) -> U. ( A X. B ) = U. (/) )
2		uni0 3771	. 2 |- U. (/) = (/)
3	1, 2	eqtrdi 2189	1 |- ( ( A X. B ) = (/) -> U. ( A X. B ) = (/) )

Syntax hints:   -> wi 4   = wceq 1332   (/)c0 3368   U.cuni 3744   X. cxp 4545

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-uni 3745

This theorem is referenced by: (None)