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Theorem mndoissmgrpOLD 35027
Description: Obsolete version of mndsgrp 17905 as of 3-Feb-2020. A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
mndoissmgrpOLD (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp)

Proof of Theorem mndoissmgrpOLD
StepHypRef Expression
1 elin 4166 . . 3 (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId ))
21simplbi 498 . 2 (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ SemiGrp)
3 df-mndo 35026 . 2 MndOp = (SemiGrp ∩ ExId )
42, 3eleq2s 2928 1 (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cin 3932   ExId cexid 35003  SemiGrpcsem 35019  MndOpcmndo 35025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940  df-mndo 35026
This theorem is referenced by:  mndoismgmOLD  35029
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