Theorem smgrpismgmOLD 38889

Description: Obsolete version of sgrpmgm 18918 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 4093	. . 3 ⊢ (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
2	1	simplbi 502	. 2 ⊢ (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma)
3		df-sgrOLD 38888	. 2 ⊢ SemiGrp = (Magma ∩ Ass)
4	2, 3	eleq2s 2878	1 ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ∩ cin 4089  Asscass 38869  Magmacmagm 38875  SemiGrpcsem 38887

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4091  df-sgrOLD 38888

This theorem is used by:  mndoismgmOLD  38897