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Mirrors > Home > MPE Home > Th. List > Mathboxes > mndoissmgrpOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mndsgrp 17905 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 4166 | . . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId )) | |
2 | 1 | simplbi 498 | . 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ SemiGrp) |
3 | df-mndo 35026 | . 2 ⊢ MndOp = (SemiGrp ∩ ExId ) | |
4 | 2, 3 | eleq2s 2928 | 1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∩ cin 3932 ExId cexid 35003 SemiGrpcsem 35019 MndOpcmndo 35025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 df-mndo 35026 |
This theorem is referenced by: mndoismgmOLD 35029 |
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