Theorem prid1 3813

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 3811	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵})
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2209  Vcvv 2821  {cpr 3706

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712

This theorem is referenced by:  prid2  3814  prnz  3831  preqr1  3888  preq12b  3890  prel12  3891  opi1  4367  opeluu  4591  onsucelsucexmidlem1  4670  regexmidlem1  4675  reg2exmidlema  4676  opthreg  4698  ordtri2or2exmid  4713  ontri2orexmidim  4714  dmrnssfld  5040  funopg  5406  acexmidlemb  6067  0lt2o  6704  2dom  7083  unfiexmid  7215  djuss  7400  exmidomni  7472  pr2cv1  7531  exmidonfinlem  7535  exmidaclem  7554  reelprrecn  8304  pnfxr  8368  sup3exmid  9277  fun2dmnop0  11280  fnpr2ob  13638  lgsdir2lem3  16063  upgrex  16258  upgr1een  16279  eulerpathprum  16635  bdop  16815  2o01f  16938  iswomni0  17006