Theorem bj-ablsscmnel 38018

Description: Abelian groups are commutative monoids (elemental version). This is a shorter proof of ablcmn 19915. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ablsscmnel	⊢ (𝐴 ∈ Abel → 𝐴 ∈ CMnd)

Proof of Theorem bj-ablsscmnel

Step	Hyp	Ref	Expression
1		bj-ablsscmn 38017	. 2 ⊢ Abel ⊆ CMnd
2	1	sseli 3930	1 ⊢ (𝐴 ∈ Abel → 𝐴 ∈ CMnd)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  CMndccmn 19908  Abelcabl 19909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-in 3909  df-ss 3919  df-abl 19911

This theorem is used by: (None)