Theorem xpexd 4864

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 4863	. 2 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> ( A X. B ) e. _V )

Syntax hints:   -> wi 4   e. wcel 2203   _Vcvv 2812   X. cxp 4746

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-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-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-opab 4171  df-xp 4754

This theorem is referenced by:  opabex2  6387  mapunen  7103  fczfsuppd  7249