Theorem elpr2 3397

Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)

Hypotheses

Ref	Expression
elpr2.1	|- B e. _V
elpr2.2	|- C e. _V

Assertion

Ref	Expression
elpr2	|- ( A e. { B , C } <-> ( A = B \/ A = C ) )

Proof of Theorem elpr2

Step	Hyp	Ref	Expression
1		elprg 3395	. . 3 |- ( A e. { B , C } -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
2	1	ibi 165	. 2 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
3		elpr2.1	. . . . . 6 |- B e. _V
4		eleq1 2100	. . . . . 6 |- ( A = B -> ( A e. _V <-> B e. _V ) )
5	3, 4	mpbiri 157	. . . . 5 |- ( A = B -> A e. _V )
6		elpr2.2	. . . . . 6 |- C e. _V
7		eleq1 2100	. . . . . 6 |- ( A = C -> ( A e. _V <-> C e. _V ) )
8	6, 7	mpbiri 157	. . . . 5 |- ( A = C -> A e. _V )
9	5, 8	jaoi 636	. . . 4 |- ( ( A = B \/ A = C ) -> A e. _V )
10		elprg 3395	. . . 4 |- ( A e. _V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
11	9, 10	syl 14	. . 3 |- ( ( A = B \/ A = C ) -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
12	11	ibir 166	. 2 |- ( ( A = B \/ A = C ) -> A e. { B , C } )
13	2, 12	impbii 117	1 |- ( A e. { B , C } <-> ( A = B \/ A = C ) )

Syntax hints:   <-> wb 98   \/ wo 629   = wceq 1243   e. wcel 1393   _Vcvv 2557   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382

This theorem is referenced by:  elxr  8696