Theorem prid1 3797

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 3795	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵})
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2203  Vcvv 2813  {cpr 3690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696

This theorem is referenced by:  prid2  3798  prnz  3815  preqr1  3872  preq12b  3874  prel12  3875  opi1  4348  opeluu  4571  onsucelsucexmidlem1  4650  regexmidlem1  4655  reg2exmidlema  4656  opthreg  4678  ordtri2or2exmid  4693  ontri2orexmidim  4694  dmrnssfld  5020  funopg  5386  acexmidlemb  6042  0lt2o  6674  2dom  7046  unfiexmid  7178  djuss  7361  exmidomni  7433  pr2cv1  7492  exmidonfinlem  7496  exmidaclem  7515  reelprrecn  8262  pnfxr  8326  sup3exmid  9231  fun2dmnop0  11222  fnpr2ob  13553  lgsdir2lem3  15903  upgrex  16098  upgr1een  16119  eulerpathprum  16475  bdop  16645  2o01f  16768  iswomni0  16836