Theorem prid1 3476

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 3474	. 2 |- ( A e. _V -> A e. { A , B } )
3	1, 2	ax-mp 7	1 |- A e. { A , B }

Syntax hints:   e. wcel 1393   _Vcvv 2557   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382

This theorem is referenced by:  prid2  3477  prnz  3490  preqr1  3539  preq12b  3541  prel12  3542  opi1  3969  opeluu  4182  onsucelsucexmidlem1  4253  regexmidlem1  4258  reg2exmidlema  4259  opthreg  4280  ordtri2or2exmid  4296  dmrnssfld  4595  funopg  4934  acexmidlemb  5504  2dom  6285  reelprrecn  7016  pnfxr  8692  bdop  9995