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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-grpssmndel | Structured version Visualization version GIF version | ||
| Description: Groups are monoids (elemental version). Shorter proof of grpmnd 18954. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bj-grpssmndel | ⊢ (𝐴 ∈ Grp → 𝐴 ∈ Mnd) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-grpssmnd 37253 | . 2 ⊢ Grp ⊆ Mnd | |
| 2 | 1 | sseli 3978 | 1 ⊢ (𝐴 ∈ Grp → 𝐴 ∈ Mnd) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Mndcmnd 18743 Grpcgrp 18947 |
| 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-grp 18950 |
| This theorem is referenced by: (None) |
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