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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-smgrpssmgmel | Structured version Visualization version GIF version |
Description: Semigroups are magmas (elemental version). (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-smgrpssmgmel | ⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-smgrpssmgm 36454 | . 2 ⊢ Smgrp ⊆ Mgm | |
2 | 1 | sseli 3979 | 1 ⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2104 Mgmcmgm 18565 Smgrpcsgrp 18645 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 df-in 3956 df-ss 3966 df-sgrp 18646 |
This theorem is referenced by: (None) |
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