Theorem prid1 3725

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 3723	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵})
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2164  Vcvv 2760  {cpr 3620

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626

This theorem is referenced by:  prid2  3726  prnz  3741  preqr1  3795  preq12b  3797  prel12  3798  opi1  4262  opeluu  4482  onsucelsucexmidlem1  4561  regexmidlem1  4566  reg2exmidlema  4567  opthreg  4589  ordtri2or2exmid  4604  ontri2orexmidim  4605  dmrnssfld  4926  funopg  5289  acexmidlemb  5911  0lt2o  6496  2dom  6861  unfiexmid  6976  djuss  7131  exmidomni  7203  exmidonfinlem  7255  exmidaclem  7270  reelprrecn  8009  pnfxr  8074  sup3exmid  8978  fnpr2ob  12926  lgsdir2lem3  15187  bdop  15437  2o01f  15557  iswomni0  15611