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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cmnssmndel | Structured version Visualization version GIF version |
Description: Commutative monoids are monoids (elemental version). This is a more direct proof of cmnmnd 18922, which relies on iscmn 18914. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cmnssmndel | ⊢ (𝐴 ∈ CMnd → 𝐴 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-cmnssmnd 34557 | . 2 ⊢ CMnd ⊆ Mnd | |
2 | 1 | sseli 3963 | 1 ⊢ (𝐴 ∈ CMnd → 𝐴 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Mndcmnd 17911 CMndccmn 18906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-in 3943 df-ss 3952 df-cmn 18908 |
This theorem is referenced by: (None) |
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