Theorem prssg

Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of Quine p. 49. (Contributed by NM, 22-Mar-2006) (Proof shortened by Andrew Salmon, 29-Jun-2011)

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

Proof

Step	Hyp	Ref	Expression
1		snssg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝐶 ↔ { 𝐴 } ⊆ 𝐶 ) )
2		snssg	⊢ ( 𝐵 ∈ 𝑊 → ( 𝐵 ∈ 𝐶 ↔ { 𝐵 } ⊆ 𝐶 ) )
3	1 2	bi2anan9	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ↔ ( { 𝐴 } ⊆ 𝐶 ∧ { 𝐵 } ⊆ 𝐶 ) ) )
4		unss	⊢ ( ( { 𝐴 } ⊆ 𝐶 ∧ { 𝐵 } ⊆ 𝐶 ) ↔ ( { 𝐴 } ∪ { 𝐵 } ) ⊆ 𝐶 )
5		df-pr	⊢ { 𝐴 , 𝐵 } = ( { 𝐴 } ∪ { 𝐵 } )
6	5	sseq1i	⊢ ( { 𝐴 , 𝐵 } ⊆ 𝐶 ↔ ( { 𝐴 } ∪ { 𝐵 } ) ⊆ 𝐶 )
7	4 6	bitr4i	⊢ ( ( { 𝐴 } ⊆ 𝐶 ∧ { 𝐵 } ⊆ 𝐶 ) ↔ { 𝐴 , 𝐵 } ⊆ 𝐶 )
8	3 7	bitrdi	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ↔ { 𝐴 , 𝐵 } ⊆ 𝐶 ) )