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Theorem smgrpismgmOLD 32831
 Description: Obsolete version of sgrpmgm 17112 as of 3-Feb-2020. A semi-group 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 3758 . . 3 (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
21simplbi 475 . 2 (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma)
3 df-sgrOLD 32830 . 2 SemiGrp = (Magma ∩ Ass)
42, 3eleq2s 2706 1 (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ∩ cin 3539  Asscass 32811  Magmacmagm 32817  SemiGrpcsem 32829 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-sgrOLD 32830 This theorem is referenced by:  mndoismgmOLD  32839
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