Theorem isabl 19439

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 19438	. 2 ⊢ Abel = (Grp ∩ CMnd)
2	1	elin2 4137	1 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∈ wcel 2104  Grpcgrp 18626  CMndccmn 19435  Abelcabl 19436

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-in 3899  df-abl 19438

This theorem is referenced by:  ablgrp  19440  ablcmn  19442  isabl2  19444  ablpropd  19446  isabld  19449  ghmabl  19483  cntrabl  19493  prdsabld  19512  unitabl  19959  tsmsinv  23348  tgptsmscls  23350  tsmsxplem1  23353  tsmsxplem2  23354  abliso  31354  gicabl  41120  2zrngaabl  45746  pgrpgt2nabl  45946