Theorem elpr2 3665

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 3663	. . 3 |- ( A e. { B , C } -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
2	1	ibi 176	. 2 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
3		elpr2.1	. . . . . 6 |- B e. _V
4		eleq1 2270	. . . . . 6 |- ( A = B -> ( A e. _V <-> B e. _V ) )
5	3, 4	mpbiri 168	. . . . 5 |- ( A = B -> A e. _V )
6		elpr2.2	. . . . . 6 |- C e. _V
7		eleq1 2270	. . . . . 6 |- ( A = C -> ( A e. _V <-> C e. _V ) )
8	6, 7	mpbiri 168	. . . . 5 |- ( A = C -> A e. _V )
9	5, 8	jaoi 718	. . . 4 |- ( ( A = B \/ A = C ) -> A e. _V )
10		elprg 3663	. . . 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 177	. 2 |- ( ( A = B \/ A = C ) -> A e. { B , C } )
13	2, 12	impbii 126	1 |- ( A e. { B , C } <-> ( A = B \/ A = C ) )

Syntax hints:   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2178   _Vcvv 2776   {cpr 3644

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650

This theorem is referenced by:  elxr  9933