Theorem isabl 18020

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 18019	. 2 ⊢ Abel = (Grp ∩ CMnd)
2	1	elin2 3763	1 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))

Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Grpcgrp 17245  CMndccmn 18016  Abelcabl 18017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-abl 18019

This theorem is referenced by:  ablgrp  18021  ablcmn  18022  isabl2  18024  ablpropd  18026  isabld  18029  ghmabl  18061  prdsabld  18088  unitabl  18491  tsmsinv  21761  tgptsmscls  21763  tsmsxplem1  21766  tsmsxplem2  21767  abliso  29027  gicabl  36687  2zrngaabl  41734  pgrpgt2nabl  41941