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	$$⊢A\in \mathrm{V}$$
	map0.2	$$⊢B\in \mathrm{V}$$
Assertion	map0	$$⊢A^B=\varnothing \leftrightarrow \left(A=\varnothing \wedge B\ne \varnothing \right)$$

Proof

Step	Hyp	Ref	Expression
1		map0.1	$$⊢A\in \mathrm{V}$$
2		map0.2	$$⊢B\in \mathrm{V}$$
3		map0g	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(A^B=\varnothing \leftrightarrow \left(A=\varnothing \wedge B\ne \varnothing \right)\right)$$
4	1 2 3	mp2an	$$⊢A^B=\varnothing \leftrightarrow \left(A=\varnothing \wedge B\ne \varnothing \right)$$