Theorem elqs 5978

Description: Membership in a quotient set. (Contributed by set.mm contributors, 23-Jul-1995.) (Revised by set.mm contributors, 12-Nov-2008.)

Hypothesis

Ref	Expression
elqs.1	⊢ B ∈ V

Assertion

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

Distinct variable groups:   x,A   x,B   x,R

Proof of Theorem elqs

Step	Hyp	Ref	Expression
1		elqs.1	. 2 ⊢ B ∈ V
2		elqsg 5977	. 2 ⊢ (B ∈ V → (B ∈ (A / R) ↔ ∃x ∈ A B = [x]R))
3	1, 2	ax-mp 5	1 ⊢ (B ∈ (A / R) ↔ ∃x ∈ A B = [x]R)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860  [cec 5946   / cqs 5947

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-rex 2621  df-v 2862  df-qs 5952

This theorem is referenced by:  qsid  5991