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Mirrors > Home > ILE Home > Th. List > elqs | GIF version |
Description: Membership in a quotient set. (Contributed by NM, 23-Jul-1995.) |
Ref | Expression |
---|---|
elqs.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
elqs | ⊢ (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqs.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | elqsg 6542 | . 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) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1342 ∈ wcel 2135 ∃wrex 2443 Vcvv 2721 [cec 6490 / cqs 6491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-qs 6498 |
This theorem is referenced by: qsss 6551 qsid 6557 erovlem 6584 nqnq0 7373 |
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