Theorem prid1 3624

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 3622	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵})
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 1480  Vcvv 2681  {cpr 3523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529

This theorem is referenced by:  prid2  3625  prnz  3640  preqr1  3690  preq12b  3692  prel12  3693  opi1  4149  opeluu  4366  onsucelsucexmidlem1  4438  regexmidlem1  4443  reg2exmidlema  4444  opthreg  4466  ordtri2or2exmid  4481  dmrnssfld  4797  funopg  5152  acexmidlemb  5759  0lt2o  6331  2dom  6692  unfiexmid  6799  djuss  6948  exmidomni  7007  exmidonfinlem  7042  exmidaclem  7057  reelprrecn  7748  pnfxr  7811  sup3exmid  8708  bdop  13062  isomninnlem  13214