Theorem elqs 8781

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

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 8780	. 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 205   = wceq 1534   ∈ wcel 2099  ∃wrex 3066  Vcvv 3470  [cec 8716   / cqs 8717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rex 3067  df-qs 8724

This theorem is referenced by:  qsss  8790  qsid  8795  erovlem  8825  sylow2blem3  19570  qusabl  19813  cldsubg  24008  qustgplem  24018  qsxpid  33068  n0elqs  37792  prter2  38347