Theorem isabl 19725

Description: The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.)

Assertion

Ref	Expression
isabl	⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))

Proof of Theorem isabl

Step	Hyp	Ref	Expression
1		df-abl 19724	. 2 ⊢ Abel = (Grp ∩ CMnd)
2	1	elin2 4157	1 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  Grpcgrp 18875  CMndccmn 19721  Abelcabl 19722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-abl 19724

This theorem is referenced by:  ablgrp  19726  ablcmn  19728  isabl2  19731  ablpropd  19733  isabld  19736  ghmabl  19773  cntrabl  19784  prdsabld  19803  unitabl  20332  tsmsinv  24104  tgptsmscls  24106  tsmsxplem1  24109  tsmsxplem2  24110  abliso  33129  primrootsunit1  42467  gicabl  43456  2zrngaabl  48610  pgrpgt2nabl  48726