Theorem isabl 19817

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 19816	. 2 ⊢ Abel = (Grp ∩ CMnd)
2	1	elin2 4213	1 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2106  Grpcgrp 18964  CMndccmn 19813  Abelcabl 19814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-in 3970  df-abl 19816

This theorem is referenced by:  ablgrp  19818  ablcmn  19820  isabl2  19823  ablpropd  19825  isabld  19828  ghmabl  19865  cntrabl  19876  prdsabld  19895  unitabl  20401  tsmsinv  24172  tgptsmscls  24174  tsmsxplem1  24177  tsmsxplem2  24178  abliso  33024  primrootsunit1  42079  gicabl  43088  2zrngaabl  48094  pgrpgt2nabl  48211