Theorem prid1 3576

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 e. _V

Assertion

Ref	Expression
prid1	|- A e. { A , B }

Proof of Theorem prid1

Step	Hyp	Ref	Expression
1		prid1.1	. 2 |- A e. _V
2		prid1g 3574	. 2 |- ( A e. _V -> A e. { A , B } )
3	1, 2	ax-mp 7	1 |- A e. { A , B }

Syntax hints:   e. wcel 1448   _Vcvv 2641   {cpr 3475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481

This theorem is referenced by:  prid2  3577  prnz  3592  preqr1  3642  preq12b  3644  prel12  3645  opi1  4092  opeluu  4309  onsucelsucexmidlem1  4381  regexmidlem1  4386  reg2exmidlema  4387  opthreg  4409  ordtri2or2exmid  4424  dmrnssfld  4738  funopg  5093  acexmidlemb  5698  0lt2o  6268  2dom  6629  unfiexmid  6735  djuss  6870  exmidomni  6926  reelprrecn  7627  pnfxr  7690  sup3exmid  8573  bdop  12654  isomninnlem  12809