Theorem unixp0im 5140

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 3798	. 2 |- ( ( A X. B ) = (/) -> U. ( A X. B ) = U. (/) )
2		uni0 3816	. 2 |- U. (/) = (/)
3	1, 2	eqtrdi 2215	1 |- ( ( A X. B ) = (/) -> U. ( A X. B ) = (/) )

Syntax hints:   -> wi 4   = wceq 1343   (/)c0 3409   U.cuni 3789   X. cxp 4602

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-uni 3790

This theorem is referenced by: (None)