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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 3800 | . 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ ExId ) | |
2 | df-mndo 33666 | . 2 ⊢ MndOp = (SemiGrp ∩ ExId ) | |
3 | 1, 2 | eleq2s 2719 | 1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∩ cin 3573 ExId cexid 33643 SemiGrpcsem 33659 MndOpcmndo 33665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-mndo 33666 |
This theorem is referenced by: mndomgmid 33670 rngo1cl 33738 |
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