Theorem sneqb 3877

Description: Biconditional equality for singletons. (Contributed by SF, 14-Jan-2015.)

Hypothesis

Ref	Expression
sneqb.1	⊢ A ∈ V

Assertion

Ref	Expression
sneqb	⊢ ({A} = {B} ↔ A = B)

Proof of Theorem sneqb

Step	Hyp	Ref	Expression
1		sneqb.1	. 2 ⊢ A ∈ V
2		sneqbg 3876	. 2 ⊢ (A ∈ V → ({A} = {B} ↔ A = B))
3	1, 2	ax-mp 5	1 ⊢ ({A} = {B} ↔ A = B)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {csn 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-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-sn 3742

This theorem is referenced by:  snelpw1  4147  otkelins2kg  4254  otkelins3kg  4255  opksnelsik  4266  nnsucelrlem1  4425  eqtfinrelk  4487  oddfinex  4505  nnadjoinlem1  4520  dfop2lem1  4574  setconslem1  4732  setconslem2  4733  funsi  5521  brsnsi  5774  brsnsi1  5776  brsnsi2  5777  funsex  5829  pw1fnex  5853  pw1fnf1o  5856  antisymex  5913  foundex  5915  extex  5916  enpw1  6063  ce2  6193  scancan  6332