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Metamath Proof Explorer

Theorem prss

Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of Quine p. 49. (Contributed by NM, 30-May-1994) (Proof shortened by Andrew Salmon, 29-Jun-2011) (Proof shortened by JJ, 23-Jul-2021)

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

Proof

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