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Mirrors > Home > MPE Home > Th. List > Mathboxes > mndoissmgrpOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mndsgrp 18766 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 3979 | . . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId )) | |
2 | 1 | simplbi 497 | . 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ SemiGrp) |
3 | df-mndo 37854 | . 2 ⊢ MndOp = (SemiGrp ∩ ExId ) | |
4 | 2, 3 | eleq2s 2857 | 1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∩ cin 3962 ExId cexid 37831 SemiGrpcsem 37847 MndOpcmndo 37853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-in 3970 df-mndo 37854 |
This theorem is referenced by: mndoismgmOLD 37857 |
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