Theorem elprg 2413

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

Assertion

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

Proof of Theorem elprg

Step	Hyp	Ref	Expression
1		eqeq1 1473	. . 3 |- (x = A -> (x = B <-> A = B))
2		eqeq1 1473	. . 3 |- (x = A -> (x = C <-> A = C))
3	1, 2	orbi12d 625	. 2 |- (x = A -> ((x = B \/ x = C) <-> (A = B \/ A = C)))
4		dfpr2 2412	. 2 |- {B, C} = {x | (x = B \/ x = C)}
5	3, 4	elab2g 1891	1 |- (A e. D -> (A e. {B, C} <-> (A = B \/ A = C)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   = wceq 953   e. wcel 955  {cpr 2400

This theorem is referenced by:  elpr 2414  elpr2 2415  ifpr 2417  elsncg 2420  pri1gOLD 2440  snsspr 2461  unisn2 2866

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403

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