Theorem smgrpismgmOLD 35300

Description: Obsolete version of sgrpmgm 17898 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 3897	. . 3 ⊢ (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
2	1	simplbi 501	. 2 ⊢ (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma)
3		df-sgrOLD 35299	. 2 ⊢ SemiGrp = (Magma ∩ Ass)
4	2, 3	eleq2s 2908	1 ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)

Syntax hints:   → wi 4   ∈ wcel 2111   ∩ cin 3880  Asscass 35280  Magmacmagm 35286  SemiGrpcsem 35298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-sgrOLD 35299

This theorem is referenced by:  mndoismgmOLD  35308