Theorem elprg 3751

Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)

Assertion

Ref	Expression
elprg	⊢ (A ∈ V → (A ∈ {B, C} ↔ (A = B ∨ A = C)))

Proof of Theorem elprg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2359	. . 3 ⊢ (x = A → (x = B ↔ A = B))
2		eqeq1 2359	. . 3 ⊢ (x = A → (x = C ↔ A = C))
3	1, 2	orbi12d 690	. 2 ⊢ (x = A → ((x = B ∨ x = C) ↔ (A = B ∨ A = C)))
4		dfpr2 3750	. 2 ⊢ {B, C} = {x ∣ (x = B ∨ x = C)}
5	3, 4	elab2g 2988	1 ⊢ (A ∈ V → (A ∈ {B, C} ↔ (A = B ∨ A = C)))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {cpr 3739

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743

This theorem is referenced by:  elpr  3752  elpr2  3753  elpri  3754  eltpg  3770  ifpr  3775  prid1g  3826