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Theorem smgrpismgmOLD 37563
Description: Obsolete version of sgrpmgm 18717 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 3963 . . 3 (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
21simplbi 496 . 2 (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma)
3 df-sgrOLD 37562 . 2 SemiGrp = (Magma ∩ Ass)
42, 3eleq2s 2844 1 (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  cin 3946  Asscass 37543  Magmacmagm 37549  SemiGrpcsem 37561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-in 3954  df-sgrOLD 37562
This theorem is referenced by:  mndoismgmOLD  37571
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