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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ablsscmn | Structured version Visualization version GIF version |
Description: Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ablsscmn | ⊢ Abel ⊆ CMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-abl 19400 | . 2 ⊢ Abel = (Grp ∩ CMnd) | |
2 | inss2 4169 | . 2 ⊢ (Grp ∩ CMnd) ⊆ CMnd | |
3 | 1, 2 | eqsstri 3960 | 1 ⊢ Abel ⊆ CMnd |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3891 ⊆ wss 3892 Grpcgrp 18588 CMndccmn 19397 Abelcabl 19398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1545 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-rab 3075 df-v 3433 df-in 3899 df-ss 3909 df-abl 19400 |
This theorem is referenced by: bj-ablsscmnel 35459 |
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