Theorem prid1g 3738

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 2206	. . 3 ⊢ 𝐴 = 𝐴
2	1	orci 733	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
3		elprg 3654	. 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 710   = wceq 1373   ∈ wcel 2177  {cpr 3635

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3171  df-sn 3640  df-pr 3641

This theorem is referenced by:  prid2g  3739  prid1  3740  preqr1g  3809  opth1  4284  en2lp  4606  acexmidlemcase  5946  pw2f1odclem  6938  en2eqpr  7011  m1expcl2  10713  maxabslemval  11563  xrmaxiflemval  11605  xrmaxaddlem  11615  2strbasg  12996  2strbas1g  12999  coseq0negpitopi  15352  structvtxval  15682