Theorem unixp0im 5280

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 3907	. 2 |- ( ( A X. B ) = (/) -> U. ( A X. B ) = U. (/) )
2		uni0 3925	. 2 |- U. (/) = (/)
3	1, 2	eqtrdi 2280	1 |- ( ( A X. B ) = (/) -> U. ( A X. B ) = (/) )

Syntax hints:   -> wi 4   = wceq 1398   (/)c0 3496   U.cuni 3898   X. cxp 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497  df-sn 3679  df-uni 3899

This theorem is referenced by: (None)