Metamath Proof Explorer


Theorem xpexb

Description: A Cartesian product exists iff its converse does. Corollary 6.9(1) in TakeutiZaring p. 26. (Contributed by Andrew Salmon, 13-Nov-2011)

Ref Expression
Assertion xpexb
|- ( ( A X. B ) e. _V <-> ( B X. A ) e. _V )

Proof

Step Hyp Ref Expression
1 cnvxp
 |-  `' ( A X. B ) = ( B X. A )
2 cnvexg
 |-  ( ( A X. B ) e. _V -> `' ( A X. B ) e. _V )
3 1 2 eqeltrrid
 |-  ( ( A X. B ) e. _V -> ( B X. A ) e. _V )
4 cnvxp
 |-  `' ( B X. A ) = ( A X. B )
5 cnvexg
 |-  ( ( B X. A ) e. _V -> `' ( B X. A ) e. _V )
6 4 5 eqeltrrid
 |-  ( ( B X. A ) e. _V -> ( A X. B ) e. _V )
7 3 6 impbii
 |-  ( ( A X. B ) e. _V <-> ( B X. A ) e. _V )