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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-mndsssmgrpel | Structured version Visualization version GIF version |
Description: Monoids are semigroups (elemental version). (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-mndsssmgrpel | ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-mndsssmgrp 36754 | . 2 ⊢ Mnd ⊆ Smgrp | |
2 | 1 | sseli 3976 | 1 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Smgrpcsgrp 18683 Mndcmnd 18699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3429 df-v 3473 df-in 3954 df-ss 3964 df-mnd 18700 |
This theorem is referenced by: (None) |
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