Theorem pri1 1841

Description: One of the two elements of an unordered pair. Part of Theorem 7.6 of [Quine] p. 49.

Hypothesis

Ref	Expression
pri1.1	⊢ A ∈ V

Assertion

Ref	Expression
pri1	⊢ A ∈ {A, B}

Proof of Theorem pri1

Step	Hyp	Ref	Expression
1		cleqid 1102	. . 3 ⊢ A = A
2		orc 225	. . 3 ⊢ (A = A → (A = A ∨ A = B))
3	1, 2	ax-mp 6	. 2 ⊢ (A = A ∨ A = B)
4		pri1.1	. . 3 ⊢ A ∈ V
5	4	elpr 1823	. 2 ⊢ (A ∈ {A, B} ↔ (A = A ∨ A = B))
6	3, 5	mpbir 165	1 ⊢ A ∈ {A, B}

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

This theorem is referenced by:  pri2 1842  tpi1 1843  prnz 1847  prss 1854  preqr1 1872  preq12b 1874  prel12 1875  opi1 1895  opth 1898  opprc1b 1906  opeluu 1953  fr2nr 2177  2dom 3332  pw2en 3348  opthreg 3455  aceq6b 3565

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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