Theorem elpri 3753

Description: If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)

Assertion

Ref	Expression
elpri	⊢ (A ∈ {B, C} → (A = B ∨ A = C))

Proof of Theorem elpri

Step	Hyp	Ref	Expression
1		elprg 3750	. 2 ⊢ (A ∈ {B, C} → (A ∈ {B, C} ↔ (A = B ∨ A = C)))
2	1	ibi 232	1 ⊢ (A ∈ {B, C} → (A = B ∨ A = C))

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:  nelpri  3754