Theorem elqs 6731

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 6730	. 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 1395   ∈ wcel 2200  ∃wrex 2509  Vcvv 2799  [cec 6676   / cqs 6677

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-qs 6684

This theorem is referenced by:  qsss  6739  qsid  6745  erovlem  6772  nqnq0  7624