Theorem elqsi 8810

Description: Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

Assertion

Ref	Expression
elqsi	⊢ (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)

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

Proof of Theorem elqsi

Step	Hyp	Ref	Expression
1		elqsg 8808	. 2 ⊢ (𝐵 ∈ (𝐴 / 𝑅) → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅))
2	1	ibi 267	1 ⊢ (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  [cec 8743   / cqs 8744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-qs 8751

This theorem is referenced by:  ectocld  8824  ecoptocl  8847  eroveu  8852  ghmqusker  19305  elrlocbasi  33270  nsgqusf1olem2  33442  qsidomlem2  33481  opprqusplusg  33517  opprqusmulr  33519  qsdrngi  33523  qsdrnglem2  33524  pstmxmet  33896