Theorem elpr 1823

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 ∈ V

Assertion

Ref	Expression
elpr	⊢ (A ∈ {B, C} ↔ (A = B ∨ A = C))

Proof of Theorem elpr

Step	Hyp	Ref	Expression
1		elpr.1	. 2 ⊢ A ∈ V
2		elprg 1822	. 2 ⊢ (A ∈ V → (A ∈ {B, C} ↔ (A = B ∨ A = C)))
3	1, 2	ax-mp 6	1 ⊢ (A ∈ {B, C} ↔ (A = B ∨ A = C))

Syntax hints:   ↔ wb 127   ∨ wo 195   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {cpr 1809

This theorem is referenced by:  hbpr 1824  eltp 1834  pri1 1841  prss 1854  prsspw 1858  preqr1 1872  preq12b 1874  prel12 1875  elop 1894  opthwiener 1914  unpr 1930  intpr 1990  fr2nr 2177  pw2en 3348

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812

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