Theorem mndoissmgrpOLD 38577

Description: Obsolete version of mndsgrp 18830 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 3922	. . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId ))
2	1	simplbi 502	. 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ SemiGrp)
3		df-mndo 38576	. 2 ⊢ MndOp = (SemiGrp ∩ ExId )
4	2, 3	eleq2s 2883	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ∩ cin 3905   ExId cexid 38553  SemiGrpcsem 38569  MndOpcmndo 38575

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-in 3913  df-mndo 38576

This theorem is used by:  mndoismgmOLD  38579