Theorem preqr1g 3753

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3755. (Contributed by Jim Kingdon, 21-Sep-2018.)

Assertion

Ref	Expression
preqr1g	|- ( ( A e. _V /\ B e. _V ) -> ( { A , C } = { B , C } -> A = B ) )

Proof of Theorem preqr1g

Step	Hyp	Ref	Expression
1		prid1g 3687	. . . . . . 7 |- ( A e. _V -> A e. { A , C } )
2		eleq2 2234	. . . . . . 7 |- ( { A , C } = { B , C } -> ( A e. { A , C } <-> A e. { B , C } ) )
3	1, 2	syl5ibcom 154	. . . . . 6 |- ( A e. _V -> ( { A , C } = { B , C } -> A e. { B , C } ) )
4		elprg 3603	. . . . . 6 |- ( A e. _V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
5	3, 4	sylibd 148	. . . . 5 |- ( A e. _V -> ( { A , C } = { B , C } -> ( A = B \/ A = C ) ) )
6	5	adantr 274	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( { A , C } = { B , C } -> ( A = B \/ A = C ) ) )
7	6	imp 123	. . 3 |- ( ( ( A e. _V /\ B e. _V ) /\ { A , C } = { B , C } ) -> ( A = B \/ A = C ) )
8		prid1g 3687	. . . . . . 7 |- ( B e. _V -> B e. { B , C } )
9		eleq2 2234	. . . . . . 7 |- ( { A , C } = { B , C } -> ( B e. { A , C } <-> B e. { B , C } ) )
10	8, 9	syl5ibrcom 156	. . . . . 6 |- ( B e. _V -> ( { A , C } = { B , C } -> B e. { A , C } ) )
11		elprg 3603	. . . . . 6 |- ( B e. _V -> ( B e. { A , C } <-> ( B = A \/ B = C ) ) )
12	10, 11	sylibd 148	. . . . 5 |- ( B e. _V -> ( { A , C } = { B , C } -> ( B = A \/ B = C ) ) )
13	12	adantl 275	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( { A , C } = { B , C } -> ( B = A \/ B = C ) ) )
14	13	imp 123	. . 3 |- ( ( ( A e. _V /\ B e. _V ) /\ { A , C } = { B , C } ) -> ( B = A \/ B = C ) )
15		eqcom 2172	. . 3 |- ( A = B <-> B = A )
16		eqeq2 2180	. . 3 |- ( A = C -> ( B = A <-> B = C ) )
17	7, 14, 15, 16	oplem1 970	. 2 |- ( ( ( A e. _V /\ B e. _V ) /\ { A , C } = { B , C } ) -> A = B )
18	17	ex 114	1 |- ( ( A e. _V /\ B e. _V ) -> ( { A , C } = { B , C } -> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 703   = wceq 1348   e. wcel 2141   _Vcvv 2730   {cpr 3584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590

This theorem is referenced by:  preqr2g  3754