Theorem preqr1g 3886

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 3888. (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 3811	. . . . . . 7 |- ( A e. _V -> A e. { A , C } )
2		eleq2 2302	. . . . . . 7 |- ( { A , C } = { B , C } -> ( A e. { A , C } <-> A e. { B , C } ) )
3	1, 2	syl5ibcom 155	. . . . . 6 |- ( A e. _V -> ( { A , C } = { B , C } -> A e. { B , C } ) )
4		elprg 3725	. . . . . 6 |- ( A e. _V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
5	3, 4	sylibd 149	. . . . 5 |- ( A e. _V -> ( { A , C } = { B , C } -> ( A = B \/ A = C ) ) )
6	5	adantr 276	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( { A , C } = { B , C } -> ( A = B \/ A = C ) ) )
7	6	imp 124	. . 3 |- ( ( ( A e. _V /\ B e. _V ) /\ { A , C } = { B , C } ) -> ( A = B \/ A = C ) )
8		prid1g 3811	. . . . . . 7 |- ( B e. _V -> B e. { B , C } )
9		eleq2 2302	. . . . . . 7 |- ( { A , C } = { B , C } -> ( B e. { A , C } <-> B e. { B , C } ) )
10	8, 9	syl5ibrcom 157	. . . . . 6 |- ( B e. _V -> ( { A , C } = { B , C } -> B e. { A , C } ) )
11		elprg 3725	. . . . . 6 |- ( B e. _V -> ( B e. { A , C } <-> ( B = A \/ B = C ) ) )
12	10, 11	sylibd 149	. . . . 5 |- ( B e. _V -> ( { A , C } = { B , C } -> ( B = A \/ B = C ) ) )
13	12	adantl 277	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( { A , C } = { B , C } -> ( B = A \/ B = C ) ) )
14	13	imp 124	. . 3 |- ( ( ( A e. _V /\ B e. _V ) /\ { A , C } = { B , C } ) -> ( B = A \/ B = C ) )
15		eqcom 2240	. . 3 |- ( A = B <-> B = A )
16		eqeq2 2248	. . 3 |- ( A = C -> ( B = A <-> B = C ) )
17	7, 14, 15, 16	oplem1 988	. 2 |- ( ( ( A e. _V /\ B e. _V ) /\ { A , C } = { B , C } ) -> A = B )
18	17	ex 115	1 |- ( ( A e. _V /\ B e. _V ) -> ( { A , C } = { B , C } -> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   _Vcvv 2821   {cpr 3706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712

This theorem is referenced by:  preqr2g  3887