Theorem elpr2 3544

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 3542	. . 3 |- ( A e. { B , C } -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
2	1	ibi 175	. 2 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
3		elpr2.1	. . . . . 6 |- B e. _V
4		eleq1 2200	. . . . . 6 |- ( A = B -> ( A e. _V <-> B e. _V ) )
5	3, 4	mpbiri 167	. . . . 5 |- ( A = B -> A e. _V )
6		elpr2.2	. . . . . 6 |- C e. _V
7		eleq1 2200	. . . . . 6 |- ( A = C -> ( A e. _V <-> C e. _V ) )
8	6, 7	mpbiri 167	. . . . 5 |- ( A = C -> A e. _V )
9	5, 8	jaoi 705	. . . 4 |- ( ( A = B \/ A = C ) -> A e. _V )
10		elprg 3542	. . . 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 176	. 2 |- ( ( A = B \/ A = C ) -> A e. { B , C } )
13	2, 12	impbii 125	1 |- ( A e. { B , C } <-> ( A = B \/ A = C ) )

Syntax hints:   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   _Vcvv 2681   {cpr 3523

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529

This theorem is referenced by:  elxr  9556