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Theorem bj-cmnssmnd 32313
 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 18018 . 2 CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g𝑥)𝑧) = (𝑧(+g𝑥)𝑦)}
2 ssrab2 3650 . 2 {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g𝑥)𝑧) = (𝑧(+g𝑥)𝑦)} ⊆ Mnd
31, 2eqsstri 3598 1 CMnd ⊆ Mnd
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∀wral 2896  {crab 2900   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Mndcmnd 17117  CMndccmn 18016 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-in 3547  df-ss 3554  df-cmn 18018 This theorem is referenced by:  bj-cmnssmndel  32314
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