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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mndoissmgrpOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of mndsgrp 18753 as of 3-Feb-2020. A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| mndoissmgrpOLD | ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elin 3967 | . . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId )) | |
| 2 | 1 | simplbi 497 | . 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ SemiGrp) | 
| 3 | df-mndo 37874 | . 2 ⊢ MndOp = (SemiGrp ∩ ExId ) | |
| 4 | 2, 3 | eleq2s 2859 | 1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ∩ cin 3950 ExId cexid 37851 SemiGrpcsem 37867 MndOpcmndo 37873 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-mndo 37874 | 
| This theorem is referenced by: mndoismgmOLD 37877 | 
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