Theorem mndoissmgrpOLD 39068

Description: Obsolete version of mndsgrp 18979 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 4096	. . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId ))
2	1	simplbi 502	. 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ SemiGrp)
3		df-mndo 39067	. 2 ⊢ MndOp = (SemiGrp ∩ ExId )
4	2, 3	eleq2s 2882	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ∩ cin 4092   ExId cexid 39044  SemiGrpcsem 39060  MndOpcmndo 39066

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-mndo 39067

This theorem is used by:  mndoismgmOLD  39070