Theorem elpr2 2429

Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15.

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 2427	. . 3 |- (A e. {B, C} -> (A e. {B, C} <-> (A = B \/ A = C)))
2	1	ibi 594	. 2 |- (A e. {B, C} -> (A = B \/ A = C))
3		elpr2.1	. . . . . 6 |- B e. V
4		eleq1 1537	. . . . . 6 |- (A = B -> (A e. V <-> B e. V))
5	3, 4	mpbiri 194	. . . . 5 |- (A = B -> A e. V)
6		elpr2.2	. . . . . 6 |- C e. V
7		eleq1 1537	. . . . . 6 |- (A = C -> (A e. V <-> C e. V))
8	6, 7	mpbiri 194	. . . . 5 |- (A = C -> A e. V)
9	5, 8	jaoi 341	. . . 4 |- ((A = B \/ A = C) -> A e. V)
10		elprg 2427	. . . 4 |- (A e. V -> (A e. {B, C} <-> (A = B \/ A = C)))
11	9, 10	syl 10	. . 3 |- ((A = B \/ A = C) -> (A e. {B, C} <-> (A = B \/ A = C)))
12	11	ibir 595	. 2 |- ((A = B \/ A = C) -> A e. {B, C})
13	2, 12	impbi 157	1 |- (A e. {B, C} <-> (A = B \/ A = C))

Syntax hints:   <-> wb 146   \/ wo 222   = wceq 958   e. wcel 960  Vcvv 1814  {cpr 2414

This theorem is referenced by:  elxr 5547

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417

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