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Theorem bj-cmnssmnd 37251
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 19796 . 2 CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g𝑥)𝑧) = (𝑧(+g𝑥)𝑦)}
21ssrab3 4081 1 CMnd ⊆ Mnd
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wral 3060  wss 3950  cfv 6559  (class class class)co 7429  Basecbs 17243  +gcplusg 17293  Mndcmnd 18743  CMndccmn 19794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-ss 3967  df-cmn 19796
This theorem is referenced by:  bj-cmnssmndel  37252
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