Metamath Proof Explorer

Theorem xpnz

Description: The Cartesian product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006)

Ref Expression
Assertion xpnz 
|- ( ( A =/= (/) /\ B =/= (/) ) <-> ( A X. B ) =/= (/) )

Proof

Step Hyp Ref Expression
1 n0 
 |-  ( A =/= (/) <-> E. x x e. A )
2 n0 
 |-  ( B =/= (/) <-> E. y y e. B )
3 1 2 anbi12i 
 |-  ( ( A =/= (/) /\ B =/= (/) ) <-> ( E. x x e. A /\ E. y y e. B ) )
4 exdistrv 
 |-  ( E. x E. y ( x e. A /\ y e. B ) <-> ( E. x x e. A /\ E. y y e. B ) )
5 3 4 bitr4i 
 |-  ( ( A =/= (/) /\ B =/= (/) ) <-> E. x E. y ( x e. A /\ y e. B ) )
6 opex 
 |-  <. x , y >. e. _V
7 eleq1 
 |-  ( z = <. x , y >. -> ( z e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
8 opelxp 
 |-  ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
9 7 8 syl6bb 
 |-  ( z = <. x , y >. -> ( z e. ( A X. B ) <-> ( x e. A /\ y e. B ) ) )
10 6 9 spcev 
 |-  ( ( x e. A /\ y e. B ) -> E. z z e. ( A X. B ) )
11 n0 
 |-  ( ( A X. B ) =/= (/) <-> E. z z e. ( A X. B ) )
12 10 11 sylibr 
 |-  ( ( x e. A /\ y e. B ) -> ( A X. B ) =/= (/) )
13 12 exlimivv 
 |-  ( E. x E. y ( x e. A /\ y e. B ) -> ( A X. B ) =/= (/) )
14 5 13 sylbi 
 |-  ( ( A =/= (/) /\ B =/= (/) ) -> ( A X. B ) =/= (/) )
15 xpeq1 
 |-  ( A = (/) -> ( A X. B ) = ( (/) X. B ) )
16 0xp 
 |-  ( (/) X. B ) = (/)
17 15 16 eqtrdi 
 |-  ( A = (/) -> ( A X. B ) = (/) )
18 17 necon3i 
 |-  ( ( A X. B ) =/= (/) -> A =/= (/) )
19 xpeq2 
 |-  ( B = (/) -> ( A X. B ) = ( A X. (/) ) )
20 xp0 
 |-  ( A X. (/) ) = (/)
21 19 20 eqtrdi 
 |-  ( B = (/) -> ( A X. B ) = (/) )
22 21 necon3i 
 |-  ( ( A X. B ) =/= (/) -> B =/= (/) )
23 18 22 jca 
 |-  ( ( A X. B ) =/= (/) -> ( A =/= (/) /\ B =/= (/) ) )
24 14 23 impbii 
 |-  ( ( A =/= (/) /\ B =/= (/) ) <-> ( A X. B ) =/= (/) )