Theorem elpr 2424

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 2423	. 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 956   e. wcel 958  Vcvv 1811  {cpr 2410

This theorem is referenced by:  hbpr 2426  ralpr 2428  eltp 2439  pri1 2450  difprsn 2465  prss 2471  prsspw 2480  preqr1 2481  preq12b 2483  prel12 2484  unipr 2515  intpr 2563  axpr 2778  zfpair2 2780  elop 2783  opthwiener 2807  fr2nr 2925  pw2en 4446  suppr 4590  ssxr 5540  unctb 7577  indistop 7648  spwpr2 8658  mapudiscn 10512  eqindhome 10541

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413

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