Theorem smgrpismgmOLD 38226

Description: Obsolete version of sgrpmgm 18686 as of 3-Feb-2020. A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
smgrpismgmOLD	⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)

Proof of Theorem smgrpismgmOLD

Step	Hyp	Ref	Expression
1		elin 3902	. . 3 ⊢ (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
2	1	simplbi 497	. 2 ⊢ (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma)
3		df-sgrOLD 38225	. 2 ⊢ SemiGrp = (Magma ∩ Ass)
4	2, 3	eleq2s 2854	1 ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)

Syntax hints:   → wi 4   ∈ wcel 2115   ∩ cin 3885  Asscass 38206  Magmacmagm 38212  SemiGrpcsem 38224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-ext 2708

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1546  df-ex 1783  df-sb 2070  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3430  df-in 3893  df-sgrOLD 38225

This theorem is referenced by:  mndoismgmOLD  38234