Theorem mndoissmgrpOLD 33667

Description: Obsolete version of mndsgrp 17299 as of 3-Feb-2020. A monoid is a semi-group. (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 3796	. . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId ))
2	1	simplbi 476	. 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ SemiGrp)
3		df-mndo 33666	. 2 ⊢ MndOp = (SemiGrp ∩ ExId )
4	2, 3	eleq2s 2719	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp)

Syntax hints:   → wi 4   ∈ wcel 1990   ∩ cin 3573   ExId cexid 33643  SemiGrpcsem 33659  MndOpcmndo 33665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-mndo 33666

This theorem is referenced by:  mndoismgmOLD  33669