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 ∈ V
	map0.2	⊢ 𝐵 ∈ V
Assertion	map0	⊢ ( ( 𝐴 ↑m 𝐵 ) = ∅ ↔ ( 𝐴 = ∅ ∧ 𝐵 ≠ ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		map0.1	⊢ 𝐴 ∈ V
2		map0.2	⊢ 𝐵 ∈ V
3		map0g	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ( 𝐴 ↑m 𝐵 ) = ∅ ↔ ( 𝐴 = ∅ ∧ 𝐵 ≠ ∅ ) ) )
4	1 2 3	mp2an	⊢ ( ( 𝐴 ↑m 𝐵 ) = ∅ ↔ ( 𝐴 = ∅ ∧ 𝐵 ≠ ∅ ) )