Metamath Proof Explorer


Theorem xpdom1

Description: Dominance law for Cartesian product. Theorem 6L(c) of Enderton p. 149. (Contributed by NM, 28-Sep-2004) (Revised by NM, 29-Mar-2006) (Revised by Mario Carneiro, 7-May-2015)

Ref Expression
Hypothesis xpdom1.2
|- C e. _V
Assertion xpdom1
|- ( A ~<_ B -> ( A X. C ) ~<_ ( B X. C ) )

Proof

Step Hyp Ref Expression
1 xpdom1.2
 |-  C e. _V
2 xpdom1g
 |-  ( ( C e. _V /\ A ~<_ B ) -> ( A X. C ) ~<_ ( B X. C ) )
3 1 2 mpan
 |-  ( A ~<_ B -> ( A X. C ) ~<_ ( B X. C ) )