Theorem prid1 3729

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 3727	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵})
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2167  Vcvv 2763  {cpr 3624

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630

This theorem is referenced by:  prid2  3730  prnz  3745  preqr1  3799  preq12b  3801  prel12  3802  opi1  4266  opeluu  4486  onsucelsucexmidlem1  4565  regexmidlem1  4570  reg2exmidlema  4571  opthreg  4593  ordtri2or2exmid  4608  ontri2orexmidim  4609  dmrnssfld  4930  funopg  5293  acexmidlemb  5915  0lt2o  6500  2dom  6865  unfiexmid  6980  djuss  7137  exmidomni  7209  exmidonfinlem  7262  exmidaclem  7277  reelprrecn  8016  pnfxr  8081  sup3exmid  8986  fnpr2ob  12993  lgsdir2lem3  15281  bdop  15531  2o01f  15651  iswomni0  15705