Theorem elqs 3227

Description: Membership in a quotient set.

Hypothesis

Ref	Expression
elqs.1	⊢ B ∈ V

Assertion

Ref	Expression
elqs	⊢ (B ∈ (A / R) ↔ ∃x(x ∈ A ∧ B = [x]R))

Distinct variable group(s):   x,A   x,B   x,R

Proof of Theorem elqs

Step	Hyp	Ref	Expression
1		elqs.1	. . 3 ⊢ B ∈ V
2		cleq1 1107	. . . 4 ⊢ (y = B → (y = [x]R ↔ B = [x]R))
3	2	birexdv 1220	. . 3 ⊢ (y = B → (∃x ∈ A y = [x]R ↔ ∃x ∈ A B = [x]R))
4		df-qs 3205	. . 3 ⊢ (A / R) = {y∣∃x ∈ A y = [x]R}
5	1, 3, 4	elab2 1419	. 2 ⊢ (B ∈ (A / R) ↔ ∃x ∈ A B = [x]R)
6		df-rex 1206	. 2 ⊢ (∃x ∈ A B = [x]R ↔ ∃x(x ∈ A ∧ B = [x]R))
7	5, 6	bitr 151	1 ⊢ (B ∈ (A / R) ↔ ∃x(x ∈ A ∧ B = [x]R))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  [cec 3198   / cqs 3199

This theorem is referenced by:  elqsi 3228  ecelqsi 3229

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-qs 3205

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