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.

Step	Hyp	Ref	Expression
1		snec.1	. . . 4 ⊢ A ∈ V
2		eceq1 5962	. . . . 5 ⊢ (x = A → [x]R = [A]R)
3	2	eqeq2d 2364	. . . 4 ⊢ (x = A → (y = [x]R ↔ y = [A]R))
4	1, 3	rexsn 3768	. . 3 ⊢ (∃x ∈ {A}y = [x]R ↔ y = [A]R)
5	4	abbii 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}
8	5, 6, 7	3eqtr4ri 2384	1 ⊢ {[A]R} = ({A} / R)

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)