Theorem xpm 4745

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 4744	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> E. w w e. ( A X. B ) )
2		eleq1 2100	. . . 4 |- ( a = x -> ( a e. A <-> x e. A ) )
3	2	cbvexv 1795	. . 3 |- ( E. a a e. A <-> E. x x e. A )
4		eleq1 2100	. . . 4 |- ( b = y -> ( b e. B <-> y e. B ) )
5	4	cbvexv 1795	. . 3 |- ( E. b b e. B <-> E. y y e. B )
6	3, 5	anbi12i 433	. 2 |- ( ( E. a a e. A /\ E. b b e. B ) <-> ( E. x x e. A /\ E. y y e. B ) )
7		eleq1 2100	. . 3 |- ( w = z -> ( w e. ( A X. B ) <-> z e. ( A X. B ) ) )
8	7	cbvexv 1795	. 2 |- ( E. w w e. ( A X. B ) <-> E. z z e. ( A X. B ) )
9	1, 6, 8	3bitr3i 199	1 |- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) )

Syntax hints:   /\ wa 97   <-> wb 98   E.wex 1381   e. wcel 1393   X. cxp 4343

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351

This theorem is referenced by:  ssxpbm  4756  xp11m  4759  xpexr2m  4762  unixpm  4853