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Theorem smgrpismgmOLD 37848
Description: Obsolete version of sgrpmgm 18749 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 3978 . . 3 (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
21simplbi 497 . 2 (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma)
3 df-sgrOLD 37847 . 2 SemiGrp = (Magma ∩ Ass)
42, 3eleq2s 2856 1 (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cin 3961  Asscass 37828  Magmacmagm 37834  SemiGrpcsem 37846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-in 3969  df-sgrOLD 37847
This theorem is referenced by:  mndoismgmOLD  37856
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