Theorem elpr 2420

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

Hypothesis

Ref	Expression
elpr.1	|- A e. V

Assertion

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

Proof of Theorem elpr

Step	Hyp	Ref	Expression
1		elpr.1	. 2 |- A e. V
2		elprg 2419	. 2 |- (A e. V -> (A e. {B, C} <-> (A = B \/ A = C)))
3	1, 2	ax-mp 7	1 |- (A e. {B, C} <-> (A = B \/ A = C))

Syntax hints:   <-> wb 146   \/ wo 222   = wceq 954   e. wcel 956  Vcvv 1807  {cpr 2406

This theorem is referenced by:  hbpr 2422  ralpr 2424  eltp 2435  pri1 2446  difprsn 2461  prss 2467  prsspw 2476  preqr1 2477  preq12b 2479  prel12 2480  unipr 2510  intpr 2558  axpr 2773  zfpair2 2775  elop 2778  opthwiener 2802  fr2nr 2920  pw2en 4432  suppr 4570  ssxr 5521  unctb 7527  indistop 7598  mapudiscn 10435

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409

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