Theorem elqs 6850

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 6849	. 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 105   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  Vcvv 2821  [cec 6795   / cqs 6796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-qs 6803

This theorem is referenced by:  qsss  6858  qsid  6864  erovlem  6891  nqnq0  7798