Theorem elpr2 3644

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 3642	. . 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 2259	. . . . . 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 2259	. . . . . 6 |- ( A = C -> ( A e. _V <-> C e. _V ) )
8	6, 7	mpbiri 168	. . . . 5 |- ( A = C -> A e. _V )
9	5, 8	jaoi 717	. . . 4 |- ( ( A = B \/ A = C ) -> A e. _V )
10		elprg 3642	. . . 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 709   = wceq 1364   e. wcel 2167   _Vcvv 2763   {cpr 3623

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629

This theorem is referenced by:  elxr  9851