Theorem isabl 19721

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 19720	. 2 ⊢ Abel = (Grp ∩ CMnd)
2	1	elin2 4169	1 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Grpcgrp 18872  CMndccmn 19717  Abelcabl 19718

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-in 3924  df-abl 19720

This theorem is referenced by:  ablgrp  19722  ablcmn  19724  isabl2  19727  ablpropd  19729  isabld  19732  ghmabl  19769  cntrabl  19780  prdsabld  19799  unitabl  20300  tsmsinv  24042  tgptsmscls  24044  tsmsxplem1  24047  tsmsxplem2  24048  abliso  32984  primrootsunit1  42092  gicabl  43095  2zrngaabl  48242  pgrpgt2nabl  48358