Theorem prid1g 3750

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 2209	. . 3 ⊢ 𝐴 = 𝐴
2	1	orci 735	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
3		elprg 3666	. 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 712   = wceq 1375   ∈ wcel 2180  {cpr 3647

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-un 3181  df-sn 3652  df-pr 3653

This theorem is referenced by:  prid2g  3751  prid1  3752  preqr1g  3823  opth1  4301  en2lp  4623  acexmidlemcase  5969  pw2f1odclem  6963  en2eqpr  7037  m1expcl2  10750  maxabslemval  11685  xrmaxiflemval  11727  xrmaxaddlem  11737  2strbasg  13119  2strbas1g  13122  coseq0negpitopi  15475  structvtxval  15805  umgrnloopv  15879  umgredgprv  15880  umgrpredgv  15910  uhgr2edg  15969  umgrvad2edg  15974