Theorem prid1 3781

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 3779	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵})
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2202  Vcvv 2803  {cpr 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680

This theorem is referenced by:  prid2  3782  prnz  3799  preqr1  3856  preq12b  3858  prel12  3859  opi1  4330  opeluu  4553  onsucelsucexmidlem1  4632  regexmidlem1  4637  reg2exmidlema  4638  opthreg  4660  ordtri2or2exmid  4675  ontri2orexmidim  4676  dmrnssfld  5001  funopg  5367  acexmidlemb  6020  0lt2o  6652  2dom  7023  unfiexmid  7153  djuss  7312  exmidomni  7384  pr2cv1  7443  exmidonfinlem  7447  exmidaclem  7466  reelprrecn  8210  pnfxr  8274  sup3exmid  9179  fun2dmnop0  11160  fnpr2ob  13486  lgsdir2lem3  15832  upgrex  16027  upgr1een  16048  eulerpathprum  16404  bdop  16574  2o01f  16697  iswomni0  16767