Theorem elqsi 8690

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 8688	. 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 1541   ∈ wcel 2111  ∃wrex 3056  [cec 8620   / cqs 8621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rex 3057  df-qs 8628

This theorem is referenced by:  ectocld  8706  ecoptocl  8731  eroveu  8736  ghmqusker  19200  elrlocbasi  33231  nsgqusf1olem2  33377  qsidomlem2  33416  opprqusplusg  33452  opprqusmulr  33454  qsdrngi  33458  qsdrnglem2  33459  pstmxmet  33908