| Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-mndsssmgrp | Structured version Visualization version GIF version | ||
| Description: Monoids are semigroups. (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bj-mndsssmgrp | ⊢ Mnd ⊆ Smgrp |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-mnd 18744 | . 2 ⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)} | |
| 2 | 1 | ssrab3 4081 | 1 ⊢ Mnd ⊆ Smgrp |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∀wral 3060 ∃wrex 3069 [wsbc 3787 ⊆ wss 3950 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 +gcplusg 17293 Smgrpcsgrp 18727 Mndcmnd 18743 |
| 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 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-ss 3967 df-mnd 18744 |
| This theorem is referenced by: bj-mndsssmgrpel 37250 |
| Copyright terms: Public domain | W3C validator |