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Theorem bj-cmnssmnd 33741
 Description: Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-cmnssmnd CMnd ⊆ Mnd

Proof of Theorem bj-cmnssmnd
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cmn 18585 . 2 CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g𝑥)𝑧) = (𝑧(+g𝑥)𝑦)}
2 ssrab2 3908 . 2 {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g𝑥)𝑧) = (𝑧(+g𝑥)𝑦)} ⊆ Mnd
31, 2eqsstri 3854 1 CMnd ⊆ Mnd
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601  ∀wral 3090  {crab 3094   ⊆ wss 3792  ‘cfv 6137  (class class class)co 6924  Basecbs 16259  +gcplusg 16342  Mndcmnd 17684  CMndccmn 18583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-in 3799  df-ss 3806  df-cmn 18585 This theorem is referenced by:  bj-cmnssmndel  33742
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