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Metamath Proof Explorer

Theorem prelpwi

Description: If two sets are members of a class, then the unordered pair of those two sets is a member of the powerclass of that class. (Contributed by Thierry Arnoux, 10-Mar-2017) (Proof shortened by AV, 23-Oct-2021)

		Ref	Expression
	Assertion	prelpwi	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → { 𝐴 , 𝐵 } ∈ 𝒫 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		prelpw	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ↔ { 𝐴 , 𝐵 } ∈ 𝒫 𝐶 ) )
2	1	ibi	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → { 𝐴 , 𝐵 } ∈ 𝒫 𝐶 )