Metamath Proof Explorer


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 ( ( 𝐴𝑉𝐵𝑊 ) → ( ( 𝐴𝐶𝐵𝐶 ) ↔ { 𝐴 , 𝐵 } ⊆ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 snssg ( 𝐴𝑉 → ( 𝐴𝐶 ↔ { 𝐴 } ⊆ 𝐶 ) )
2 snssg ( 𝐵𝑊 → ( 𝐵𝐶 ↔ { 𝐵 } ⊆ 𝐶 ) )
3 1 2 bi2anan9 ( ( 𝐴𝑉𝐵𝑊 ) → ( ( 𝐴𝐶𝐵𝐶 ) ↔ ( { 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 ( ( 𝐴𝑉𝐵𝑊 ) → ( ( 𝐴𝐶𝐵𝐶 ) ↔ { 𝐴 , 𝐵 } ⊆ 𝐶 ) )