Theorem prsspwg

Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016) (Revised by NM, 18-Jan-2018)

	Ref	Expression
Assertion	prsspwg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( { 𝐴 , 𝐵 } ⊆ 𝒫 𝐶 ↔ ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		prssg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐴 ∈ 𝒫 𝐶 ∧ 𝐵 ∈ 𝒫 𝐶 ) ↔ { 𝐴 , 𝐵 } ⊆ 𝒫 𝐶 ) )
2		elpwg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝒫 𝐶 ↔ 𝐴 ⊆ 𝐶 ) )
3		elpwg	⊢ ( 𝐵 ∈ 𝑊 → ( 𝐵 ∈ 𝒫 𝐶 ↔ 𝐵 ⊆ 𝐶 ) )
4	2 3	bi2anan9	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐴 ∈ 𝒫 𝐶 ∧ 𝐵 ∈ 𝒫 𝐶 ) ↔ ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) ) )
5	1 4	bitr3d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( { 𝐴 , 𝐵 } ⊆ 𝒫 𝐶 ↔ ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) ) )