Theorem xpexd 4890

Description: The Cartesian product of two sets is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
xpexd.1	|- ( ph -> A e. V )
xpexd.2	|- ( ph -> B e. W )

Assertion

Ref	Expression
xpexd	|- ( ph -> ( A X. B ) e. _V )

Proof of Theorem xpexd

Step	Hyp	Ref	Expression
1		xpexd.1	. 2 |- ( ph -> A e. V )
2		xpexd.2	. 2 |- ( ph -> B e. W )
3		xpexg 4889	. 2 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )
4	1, 2, 3	syl2anc 415	1 |- ( ph -> ( A X. B ) e. _V )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   _Vcvv 2821   X. cxp 4772

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  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-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-opab 4193  df-xp 4780

This theorem is used by:  opabex2  6428  mapunen  7151  fczfsuppd  7297  aprprop  14603