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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 19389 | . 2 ⊢ Abel = (Grp ∩ CMnd) | |
2 | inss2 4163 | . 2 ⊢ (Grp ∩ CMnd) ⊆ CMnd | |
3 | 1, 2 | eqsstri 3955 | 1 ⊢ Abel ⊆ CMnd |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3886 ⊆ wss 3887 Grpcgrp 18577 CMndccmn 19386 Abelcabl 19387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-in 3894 df-ss 3904 df-abl 19389 |
This theorem is referenced by: bj-ablsscmnel 35450 |
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