Theorem elqs 8701

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 8700	. 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 207   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  Vcvv 3431  [cec 8631   / cqs 8632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rex 3064  df-qs 8639

This theorem is referenced by:  qsss  8712  qsid  8718  erovlem  8750  sylow2blem3  19588  qusabl  19831  cldsubg  24094  qustgplem  24104  qsxpid  33445  n0elqs  38699  prter2  39373