Theorem prid1 3777

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 3775	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵})
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2202  Vcvv 2802  {cpr 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by:  prid2  3778  prnz  3795  preqr1  3851  preq12b  3853  prel12  3854  opi1  4324  opeluu  4547  onsucelsucexmidlem1  4626  regexmidlem1  4631  reg2exmidlema  4632  opthreg  4654  ordtri2or2exmid  4669  ontri2orexmidim  4670  dmrnssfld  4995  funopg  5360  acexmidlemb  6010  0lt2o  6609  2dom  6980  unfiexmid  7110  djuss  7269  exmidomni  7341  pr2cv1  7400  exmidonfinlem  7404  exmidaclem  7423  reelprrecn  8167  pnfxr  8232  sup3exmid  9137  fun2dmnop0  11115  fnpr2ob  13428  lgsdir2lem3  15765  upgrex  15960  upgr1een  15981  eulerpathprum  16337  bdop  16496  2o01f  16619  iswomni0  16682