Metamath Proof Explorer


Theorem map0

Description: Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of Suppes p. 89. (Contributed by NM, 10-Dec-2003)

Ref Expression
Hypotheses map0.1
|- A e. _V
map0.2
|- B e. _V
Assertion map0
|- ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) )

Proof

Step Hyp Ref Expression
1 map0.1
 |-  A e. _V
2 map0.2
 |-  B e. _V
3 map0g
 |-  ( ( A e. _V /\ B e. _V ) -> ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) ) )
4 1 2 3 mp2an
 |-  ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) )