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Theorem bj-cmnssmnd 33266
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 18241 . 2 CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g𝑥)𝑧) = (𝑧(+g𝑥)𝑦)}
2 ssrab2 3720 . 2 {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g𝑥)𝑧) = (𝑧(+g𝑥)𝑦)} ⊆ Mnd
31, 2eqsstri 3668 1 CMnd ⊆ Mnd
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wral 2941  {crab 2945  wss 3607  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Mndcmnd 17341  CMndccmn 18239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-in 3614  df-ss 3621  df-cmn 18241
This theorem is referenced by:  bj-cmnssmndel  33267
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