Theorem prid1 3667

Description: An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
prid1.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prid1	⊢ 𝐴 ∈ {𝐴, 𝐵}

Proof of Theorem prid1

Step	Hyp	Ref	Expression
1		prid1.1	. 2 ⊢ 𝐴 ∈ V
2		prid1g 3665	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵})
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2128  Vcvv 2712  {cpr 3562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3567  df-pr 3568

This theorem is referenced by:  prid2  3668  prnz  3683  preqr1  3733  preq12b  3735  prel12  3736  opi1  4194  opeluu  4412  onsucelsucexmidlem1  4489  regexmidlem1  4494  reg2exmidlema  4495  opthreg  4517  ordtri2or2exmid  4532  ontri2orexmidim  4533  dmrnssfld  4851  funopg  5206  acexmidlemb  5818  0lt2o  6390  2dom  6752  unfiexmid  6864  djuss  7016  exmidomni  7087  exmidonfinlem  7130  exmidaclem  7145  reelprrecn  7869  pnfxr  7932  sup3exmid  8833  bdop  13521  2o01f  13639  iswomni0  13693