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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 8557 | . 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 205 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 Vcvv 3432 [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: qsss 8567 qsid 8572 erovlem 8602 sylow2blem3 19227 qusabl 19466 cldsubg 23262 qustgplem 23272 qsxpid 31558 n0elqs 36461 prter2 36895 |
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