Theorem mndoissmgrpOLD 38519

Description: Obsolete version of mndsgrp 18793 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

Step	Hyp	Ref	Expression
1		elin 3921	. . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId ))
2	1	simplbi 501	. 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ SemiGrp)
3		df-mndo 38518	. 2 ⊢ MndOp = (SemiGrp ∩ ExId )
4	2, 3	eleq2s 2881	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp)

Syntax hints:   → wi 4   ∈ wcel 2143   ∩ cin 3904   ExId cexid 38495  SemiGrpcsem 38511  MndOpcmndo 38517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-in 3912  df-mndo 38518

This theorem is referenced by:  mndoismgmOLD  38521