Theorem prid1g 3674

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 2164	. . 3 ⊢ 𝐴 = 𝐴
2	1	orci 721	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
3		elprg 3590	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴, 𝐵} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)))
4	2, 3	mpbiri 167	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∨ wo 698   = wceq 1342   ∈ wcel 2135  {cpr 3571

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577

This theorem is referenced by:  prid2g  3675  prid1  3676  preqr1g  3740  opth1  4208  en2lp  4525  acexmidlemcase  5831  en2eqpr  6864  m1expcl2  10467  maxabslemval  11136  xrmaxiflemval  11177  xrmaxaddlem  11187  2strbasg  12432  2strbas1g  12435  coseq0negpitopi  13298