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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ablsscmnel | Structured version Visualization version GIF version |
Description: Abelian groups are commutative monoids (elemental version). This is a shorter proof of ablcmn 19703. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ablsscmnel | ⊢ (𝐴 ∈ Abel → 𝐴 ∈ CMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-ablsscmn 36622 | . 2 ⊢ Abel ⊆ CMnd | |
2 | 1 | sseli 3978 | 1 ⊢ (𝐴 ∈ Abel → 𝐴 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 CMndccmn 19696 Abelcabl 19697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-in 3955 df-ss 3965 df-abl 19699 |
This theorem is referenced by: (None) |
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