Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 8379 | . 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 209 = wceq 1542 ∈ wcel 2114 ∃wrex 3054 Vcvv 3398 [cec 8318 / cqs 8319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-rex 3059 df-qs 8326 |
This theorem is referenced by: qsss 8389 qsid 8394 erovlem 8424 sylow2blem3 18865 qusabl 19104 cldsubg 22862 qustgplem 22872 qsxpid 31130 n0elqs 36084 prter2 36518 |
Copyright terms: Public domain | W3C validator |