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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 18208, which relies on iscmn 18200. (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 33136 | . 2 ⊢ CMnd ⊆ Mnd | |
2 | 1 | sseli 3599 | 1 ⊢ (𝐴 ∈ CMnd → 𝐴 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 Mndcmnd 17294 CMndccmn 18193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-in 3581 df-ss 3588 df-cmn 18195 |
This theorem is referenced by: (None) |
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