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 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