Theorem elprg 2468

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 1518	. . 3 |- (x = A -> (x = B <-> A = B))
2		eqeq1 1518	. . 3 |- (x = A -> (x = C <-> A = C))
3	1, 2	orbi12d 629	. 2 |- (x = A -> ((x = B \/ x = C) <-> (A = B \/ A = C)))
4		dfpr2 2467	. 2 |- {B, C} = {x | (x = B \/ x = C)}
5	3, 4	elab2g 1938	1 |- (A e. D -> (A e. {B, C} <-> (A = B \/ A = C)))

Syntax hints:   -> wi 3   <-> wb 144   \/ wo 220   = wceq 988   e. wcel 990  {cpr 2455

This theorem is referenced by:  elpr 2469  elpr2 2470  ifpr 2472  elsncg 2475  prid1g 2495  snsspr 2518  unisn2 2929

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-12 1000  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-clab 1500  df-cleq 1505  df-clel 1508  df-v 1850  df-un 2094  df-sn 2457  df-pr 2458

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