| 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 8808 | . 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 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 [cec 8743 / cqs 8744 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-qs 8751 |
| This theorem is referenced by: qsss 8818 qsid 8823 erovlem 8853 sylow2blem3 19640 qusabl 19883 cldsubg 24119 qustgplem 24129 qsxpid 33390 n0elqs 38327 prter2 38882 |
| Copyright terms: Public domain | W3C validator |