Theorem prid1g 3825

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 2353	. . 3 ⊢ A = A
2	1	orci 379	. 2 ⊢ (A = A ∨ A = B)
3		elprg 3750	. 2 ⊢ (A ∈ V → (A ∈ {A, B} ↔ (A = A ∨ A = B)))
4	2, 3	mpbiri 224	1 ⊢ (A ∈ V → A ∈ {A, B})

Syntax hints:   → wi 4   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {cpr 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742

This theorem is referenced by:  prid2g  3826  prid1  3827  opkth1g  4130