Theorem prid1 3739

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 3737	. 2 |- ( A e. _V -> A e. { A , B } )
3	1, 2	ax-mp 5	1 |- A e. { A , B }

Syntax hints:   e. wcel 2176   _Vcvv 2772   {cpr 3634

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640

This theorem is referenced by:  prid2  3740  prnz  3755  preqr1  3809  preq12b  3811  prel12  3812  opi1  4277  opeluu  4498  onsucelsucexmidlem1  4577  regexmidlem1  4582  reg2exmidlema  4583  opthreg  4605  ordtri2or2exmid  4620  ontri2orexmidim  4621  dmrnssfld  4942  funopg  5306  acexmidlemb  5938  0lt2o  6529  2dom  6899  unfiexmid  7017  djuss  7174  exmidomni  7246  exmidonfinlem  7303  exmidaclem  7322  reelprrecn  8062  pnfxr  8127  sup3exmid  9032  fun2dmnop0  10994  fnpr2ob  13205  lgsdir2lem3  15540  bdop  15848  2o01f  15968  iswomni0  16027