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Theorem smgrpismgmOLD 33993
 Description: Obsolete version of sgrpmgm 17497 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 3947 . . 3 (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
21simplbi 485 . 2 (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma)
3 df-sgrOLD 33992 . 2 SemiGrp = (Magma ∩ Ass)
42, 3eleq2s 2868 1 (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2145   ∩ cin 3722  Asscass 33973  Magmacmagm 33979  SemiGrpcsem 33991 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-sgrOLD 33992 This theorem is referenced by:  mndoismgmOLD  34001
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