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Mirrors > Home > MPE Home > Th. List > Mathboxes > mndoissmgrpOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mndsgrp 18570 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 3930 | . . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId )) | |
2 | 1 | simplbi 499 | . 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ SemiGrp) |
3 | df-mndo 36376 | . 2 ⊢ MndOp = (SemiGrp ∩ ExId ) | |
4 | 2, 3 | eleq2s 2852 | 1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∩ cin 3913 ExId cexid 36353 SemiGrpcsem 36369 MndOpcmndo 36375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3449 df-in 3921 df-mndo 36376 |
This theorem is referenced by: mndoismgmOLD 36379 |
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