Theorem elqsi 8559

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 8557	. 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 2106  ∃wrex 3065  [cec 8496   / cqs 8497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-qs 8504

This theorem is referenced by:  ectocld  8573  ecoptocl  8596  eroveu  8601  nsgqusf1olem2  31599  qsidomlem2  31629  pstmxmet  31847