Metamath Proof Explorer


Theorem elpr2g

Description: A member of a pair of sets is one or the other of them, and conversely. Exercise 1 of TakeutiZaring p. 15. (Contributed by NM, 14-Oct-2005) Generalize from sethood hypothesis to sethood antecedent. (Revised by BJ, 25-May-2024)

Ref Expression
Assertion elpr2g B V C W A B C A = B A = C

Proof

Step Hyp Ref Expression
1 elex A B C A V
2 1 a1i B V C W A B C A V
3 elex B V B V
4 eleq1a B V A = B A V
5 3 4 syl B V A = B A V
6 elex C W C V
7 eleq1a C V A = C A V
8 6 7 syl C W A = C A V
9 5 8 jaao B V C W A = B A = C A V
10 elprg A V A B C A = B A = C
11 10 a1i B V C W A V A B C A = B A = C
12 2 9 11 pm5.21ndd B V C W A B C A = B A = C