Theorem altopelaltxp 32083

Description: Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5146, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)

Assertion

Ref	Expression
altopelaltxp	|- ( << X , Y >> e. ( A XX. B ) <-> ( X e. A /\ Y e. B ) )

Proof of Theorem altopelaltxp

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elaltxp 32082	. 2 |- ( << X , Y >> e. ( A XX. B ) <-> E. x e. A E. y e. B << X , Y >> = << x , y >> )
2		reeanv 3107	. . 3 |- ( E. x e. A E. y e. B ( x = X /\ y = Y ) <-> ( E. x e. A x = X /\ E. y e. B y = Y ) )
3		eqcom 2629	. . . . 5 |- ( << X , Y >> = << x , y >> <-> << x , y >> = << X , Y >> )
4		vex 3203	. . . . . 6 |- x e. _V
5		vex 3203	. . . . . 6 |- y e. _V
6	4, 5	altopth 32076	. . . . 5 |- ( << x , y >> = << X , Y >> <-> ( x = X /\ y = Y ) )
7	3, 6	bitri 264	. . . 4 |- ( << X , Y >> = << x , y >> <-> ( x = X /\ y = Y ) )
8	7	2rexbii 3042	. . 3 |- ( E. x e. A E. y e. B << X , Y >> = << x , y >> <-> E. x e. A E. y e. B ( x = X /\ y = Y ) )
9		risset 3062	. . . 4 |- ( X e. A <-> E. x e. A x = X )
10		risset 3062	. . . 4 |- ( Y e. B <-> E. y e. B y = Y )
11	9, 10	anbi12i 733	. . 3 |- ( ( X e. A /\ Y e. B ) <-> ( E. x e. A x = X /\ E. y e. B y = Y ) )
12	2, 8, 11	3bitr4i 292	. 2 |- ( E. x e. A E. y e. B << X , Y >> = << x , y >> <-> ( X e. A /\ Y e. B ) )
13	1, 12	bitri 264	1 |- ( << X , Y >> e. ( A XX. B ) <-> ( X e. A /\ Y e. B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   <<caltop 32063   XX. caltxp 32064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-altop 32065  df-altxp 32066

This theorem is referenced by: (None)