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Theorem smgrpismgmOLD 35200
Description: Obsolete version of sgrpmgm 17895 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
StepHypRef Expression
1 elin 3934 . . 3 (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
21simplbi 501 . 2 (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma)
3 df-sgrOLD 35199 . 2 SemiGrp = (Magma ∩ Ass)
42, 3eleq2s 2934 1 (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  cin 3917  Asscass 35180  Magmacmagm 35186  SemiGrpcsem 35198
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3481  df-in 3925  df-sgrOLD 35199
This theorem is referenced by:  mndoismgmOLD  35208
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