Theorem prid1 3772

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 3770	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵})
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2200  Vcvv 2799  {cpr 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  prid2  3773  prnz  3790  preqr1  3846  preq12b  3848  prel12  3849  opi1  4318  opeluu  4541  onsucelsucexmidlem1  4620  regexmidlem1  4625  reg2exmidlema  4626  opthreg  4648  ordtri2or2exmid  4663  ontri2orexmidim  4664  dmrnssfld  4987  funopg  5352  acexmidlemb  5999  0lt2o  6595  2dom  6966  unfiexmid  7091  djuss  7248  exmidomni  7320  pr2cv1  7379  exmidonfinlem  7382  exmidaclem  7401  reelprrecn  8145  pnfxr  8210  sup3exmid  9115  fun2dmnop0  11082  fnpr2ob  13388  lgsdir2lem3  15724  upgrex  15918  bdop  16293  2o01f  16417  iswomni0  16479