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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 18916. (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 34564 | . 2 ⊢ Abel ⊆ CMnd | |
2 | 1 | sseli 3966 | 1 ⊢ (𝐴 ∈ Abel → 𝐴 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 CMndccmn 18909 Abelcabl 18910 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-in 3946 df-ss 3955 df-abl 18912 |
This theorem is referenced by: (None) |
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