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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.

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)

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-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-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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