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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 18907. (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 34554 | . 2 ⊢ Abel ⊆ CMnd | |
2 | 1 | sseli 3962 | 1 ⊢ (𝐴 ∈ Abel → 𝐴 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 CMndccmn 18900 Abelcabl 18901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 df-abl 18903 |
This theorem is referenced by: (None) |
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