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 17907 | . 2 ⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)} | |
2 | 1 | ssrab3 4050 | 1 ⊢ Mnd ⊆ Smgrp |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∀wral 3137 ∃wrex 3138 [wsbc 3768 ⊆ wss 3929 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 +gcplusg 16560 Smgrpcsgrp 17895 Mndcmnd 17906 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-in 3936 df-ss 3945 df-mnd 17907 |
This theorem is referenced by: bj-mndsssmgrpel 34577 |
Copyright terms: Public domain | W3C validator |