Theorem prid1g 3814

Description: An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Assertion

Ref	Expression
prid1g	⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})

Proof of Theorem prid1g

Step	Hyp	Ref	Expression
1		eqid 2238	. . 3 ⊢ 𝐴 = 𝐴
2	1	orci 743	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
3		elprg 3728	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴, 𝐵} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)))
4	2, 3	mpbiri 168	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∨ wo 720   = wceq 1402   ∈ wcel 2209  {cpr 3709

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715

This theorem is referenced by:  prid2g  3815  prid1  3816  preqr1g  3889  opth1  4374  en2lp  4699  acexmidlemcase  6074  pw2f1odclem  7128  en2eqpr  7208  m1expcl2  10981  maxabslemval  11957  xrmaxiflemval  11999  xrmaxaddlem  12009  2strbasg  13457  2strbas1g  13460  coseq0negpitopi  15920  structvtxval  16263  umgrnloopv  16338  umgredgprv  16339  umgrpredgv  16371  uhgr2edg  16430  umgrvad2edg  16435  usgr2v1e2w  16470  1hegrvtxdg1fi  16533  vdegp1bid  16539