Theorem smgrpismgmOLD 35139

Description: Obsolete version of sgrpmgm 17905 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 4168	. . 3 ⊢ (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
2	1	simplbi 500	. 2 ⊢ (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma)
3		df-sgrOLD 35138	. 2 ⊢ SemiGrp = (Magma ∩ Ass)
4	2, 3	eleq2s 2931	1 ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)

Syntax hints:   → wi 4   ∈ wcel 2110   ∩ cin 3934  Asscass 35119  Magmacmagm 35125  SemiGrpcsem 35137

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3942  df-sgrOLD 35138

This theorem is referenced by:  mndoismgmOLD  35147