Theorem unixp0im 5001

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 3684	. 2 |- ( ( A X. B ) = (/) -> U. ( A X. B ) = U. (/) )
2		uni0 3702	. 2 |- U. (/) = (/)
3	1, 2	syl6eq 2143	1 |- ( ( A X. B ) = (/) -> U. ( A X. B ) = (/) )

Syntax hints:   -> wi 4   = wceq 1296   (/)c0 3302   U.cuni 3675   X. cxp 4465

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-in 3019  df-ss 3026  df-nul 3303  df-sn 3472  df-uni 3676

This theorem is referenced by: (None)