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Theorem snec 5987
 Description: The singleton of an equivalence class. (Contributed by set.mm contributors, 29-Jan-1999.) (Revised by set.mm contributors, 9-Jul-2014.)
Hypothesis
Ref Expression
snec.1 A V
Assertion
Ref Expression
snec {[A]R} = ({A} / R)

Proof of Theorem snec
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snec.1 . . . 4 A V
2 eceq1 5962 . . . . 5 (x = A → [x]R = [A]R)
32eqeq2d 2364 . . . 4 (x = A → (y = [x]Ry = [A]R))
41, 3rexsn 3768 . . 3 (x {A}y = [x]Ry = [A]R)
54abbii 2465 . 2 {y x {A}y = [x]R} = {y y = [A]R}
6 df-qs 5951 . 2 ({A} / R) = {y x {A}y = [x]R}
7 df-sn 3741 . 2 {[A]R} = {y y = [A]R}
85, 6, 73eqtr4ri 2384 1 {[A]R} = ({A} / R)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859  {csn 3737  [cec 5945   / cqs 5946 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-3an 936  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-rex 2620  df-v 2861  df-sbc 3047  df-sn 3741  df-ima 4727  df-ec 5947  df-qs 5951 This theorem is referenced by: (None)
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