Theorem elpr 3751

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

Hypothesis

Ref	Expression
elpr.1	⊢ A ∈ V

Assertion

Ref	Expression
elpr	⊢ (A ∈ {B, C} ↔ (A = B ∨ A = C))

Proof of Theorem elpr

Step	Hyp	Ref	Expression
1		elpr.1	. 2 ⊢ A ∈ V
2		elprg 3750	. 2 ⊢ (A ∈ V → (A ∈ {B, C} ↔ (A = B ∨ A = C)))
3	1, 2	ax-mp 8	1 ⊢ (A ∈ {B, C} ↔ (A = B ∨ A = C))

Syntax hints:   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {cpr 3738

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  difprsnss  3846  pwpr  3883  pwtp  3884  unipr  3905  intpr  3959  axprimlem2  4089  preqr1  4124  preq12b  4127  enprmaplem3  6078  enprmaplem5  6080  2p1e3c  6156