This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem prsspw

Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003) (Proof shortened by Andrew Salmon, 26-Jun-2011) (Proof shortened by OpenAI, 25-Mar-2020)

	Ref	Expression
Hypotheses	prsspw.1	⊢ 𝐴 ∈ V
	prsspw.2	⊢ 𝐵 ∈ V
Assertion	prsspw	⊢ ( { 𝐴 , 𝐵 } ⊆ 𝒫 𝐶 ↔ ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		prsspw.1	⊢ 𝐴 ∈ V
2		prsspw.2	⊢ 𝐵 ∈ V
3		prsspwg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( { 𝐴 , 𝐵 } ⊆ 𝒫 𝐶 ↔ ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) ) )
4	1 2 3	mp2an	⊢ ( { 𝐴 , 𝐵 } ⊆ 𝒫 𝐶 ↔ ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) )