Theorem prid1 3724

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 3722	. 2 |- ( A e. _V -> A e. { A , B } )
3	1, 2	ax-mp 5	1 |- A e. { A , B }

Syntax hints:   e. wcel 2164   _Vcvv 2760   {cpr 3619

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625

This theorem is referenced by:  prid2  3725  prnz  3740  preqr1  3794  preq12b  3796  prel12  3797  opi1  4261  opeluu  4481  onsucelsucexmidlem1  4560  regexmidlem1  4565  reg2exmidlema  4566  opthreg  4588  ordtri2or2exmid  4603  ontri2orexmidim  4604  dmrnssfld  4925  funopg  5288  acexmidlemb  5910  0lt2o  6494  2dom  6859  unfiexmid  6974  djuss  7129  exmidomni  7201  exmidonfinlem  7253  exmidaclem  7268  reelprrecn  8007  pnfxr  8072  sup3exmid  8976  fnpr2ob  12923  lgsdir2lem3  15146  bdop  15367  2o01f  15487  iswomni0  15541