Theorem elqsi 8517

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 8515	. 2 ⊢ (𝐵 ∈ (𝐴 / 𝑅) → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅))
2	1	ibi 266	1 ⊢ (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  [cec 8454   / cqs 8455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rex 3069  df-qs 8462

This theorem is referenced by:  ectocld  8531  ecoptocl  8554  eroveu  8559  nsgqusf1olem2  31501  qsidomlem2  31531  pstmxmet  31749