Theorem mndoissmgrpOLD 37869

Description: Obsolete version of mndsgrp 18674 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 3933	. . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId ))
2	1	simplbi 497	. 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ SemiGrp)
3		df-mndo 37868	. 2 ⊢ MndOp = (SemiGrp ∩ ExId )
4	2, 3	eleq2s 2847	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp)

Syntax hints:   → wi 4   ∈ wcel 2109   ∩ cin 3916   ExId cexid 37845  SemiGrpcsem 37861  MndOpcmndo 37867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-in 3924  df-mndo 37868

This theorem is referenced by:  mndoismgmOLD  37871