Theorem xpm 4960

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 4959	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. w w e. ( A X. B ) )
2		eleq1 2202	. . . 4 |- ( a = x -> ( a e. A <-> x e. A ) )
3	2	cbvexv 1890	. . 3 |- ( E. a a e. A <-> E. x x e. A )
4		eleq1 2202	. . . 4 |- ( b = y -> ( b e. B <-> y e. B ) )
5	4	cbvexv 1890	. . 3 |- ( E. b b e. B <-> E. y y e. B )
6	3, 5	anbi12i 455	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> ( E. x x e. A /\ E. y y e. B ) )
7		eleq1 2202	. . 3 |- ( w = z -> ( w e. ( A X. B ) <-> z e. ( A X. B ) ) )
8	7	cbvexv 1890	. 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 1468   e. wcel 1480   X. cxp 4537

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545

This theorem is referenced by:  ssxpbm  4974  xp11m  4977  xpexr2m  4980  unixpm  5074