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Mirrors > Home > MPE Home > Th. List > Mathboxes > mndoissmgrpOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mndsgrp 17919 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 4171 | . . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId )) | |
2 | 1 | simplbi 500 | . 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ SemiGrp) |
3 | df-mndo 35147 | . 2 ⊢ MndOp = (SemiGrp ∩ ExId ) | |
4 | 2, 3 | eleq2s 2933 | 1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∩ cin 3937 ExId cexid 35124 SemiGrpcsem 35140 MndOpcmndo 35146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-in 3945 df-mndo 35147 |
This theorem is referenced by: mndoismgmOLD 35150 |
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