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Mirrors > Home > MPE Home > Th. List > elqs | Structured version Visualization version 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 7685 | . 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 setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 [cec 7627 / cqs 7628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-qs 7635 |
This theorem is referenced by: qsss 7695 qsid 7700 erovlem 7730 sylow2blem3 17860 qusabl 18091 cldsubg 21724 qustgplem 21734 prter2 33184 |
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