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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 18901 | . 2 ⊢ Abel = (Grp ∩ CMnd) | |
2 | inss2 4204 | . 2 ⊢ (Grp ∩ CMnd) ⊆ CMnd | |
3 | 1, 2 | eqsstri 3999 | 1 ⊢ Abel ⊆ CMnd |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3933 ⊆ wss 3934 Grpcgrp 18095 CMndccmn 18898 Abelcabl 18899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-rab 3145 df-v 3495 df-in 3941 df-ss 3950 df-abl 18901 |
This theorem is referenced by: bj-ablsscmnel 34553 |
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