Theorem xpm 4968

Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)

Assertion

Ref	Expression
xpm	|- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) )

Distinct variable groups:   x, A   y, B   z, A   z, B

Allowed substitution hints:   A( y)   B( x)

Proof of Theorem xpm

Dummy variables a b w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpmlem 4967	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. w w e. ( A X. B ) )
2		eleq1 2203	. . . 4 |- ( a = x -> ( a e. A <-> x e. A ) )
3	2	cbvexv 1891	. . 3 |- ( E. a a e. A <-> E. x x e. A )
4		eleq1 2203	. . . 4 |- ( b = y -> ( b e. B <-> y e. B ) )
5	4	cbvexv 1891	. . 3 |- ( E. b b e. B <-> E. y y e. B )
6	3, 5	anbi12i 456	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> ( E. x x e. A /\ E. y y e. B ) )
7		eleq1 2203	. . 3 |- ( w = z -> ( w e. ( A X. B ) <-> z e. ( A X. B ) ) )
8	7	cbvexv 1891	. 2 |- ( E. w w e. ( A X. B ) <-> E. z z e. ( A X. B ) )
9	1, 6, 8	3bitr3i 209	1 |- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1469   e. wcel 1481   X. cxp 4545

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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-opab 3998  df-xp 4553

This theorem is referenced by:  ssxpbm  4982  xp11m  4985  xpexr2m  4988  unixpm  5082