Theorem unixp0im 5322

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 3942	. 2 |- ( ( A X. B ) = (/) -> U. ( A X. B ) = U. (/) )
2		uni0 3960	. 2 |- U. (/) = (/)
3	1, 2	eqtrdi 2287	1 |- ( ( A X. B ) = (/) -> U. ( A X. B ) = (/) )

Syntax hints:   -> wi 4   = wceq 1402   (/)c0 3520   U.cuni 3933   X. cxp 4770

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3714  df-uni 3934

This theorem is referenced by: (None)