Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > mndoisexid | Structured version Visualization version GIF version |
Description: A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mndoisexid | ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elinel2 4173 | . 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ ExId ) | |
2 | df-mndo 35160 | . 2 ⊢ MndOp = (SemiGrp ∩ ExId ) | |
3 | 1, 2 | eleq2s 2931 | 1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∩ cin 3935 ExId cexid 35137 SemiGrpcsem 35153 MndOpcmndo 35159 |
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 2793 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 df-mndo 35160 |
This theorem is referenced by: mndomgmid 35164 rngo1cl 35232 |
Copyright terms: Public domain | W3C validator |