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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 | elin 3758 | . . 3 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId )) | |
2 | 1 | simprbi 479 | . 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ ExId ) |
3 | df-mndo 32836 | . 2 ⊢ MndOp = (SemiGrp ∩ ExId ) | |
4 | 2, 3 | eleq2s 2706 | 1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∩ cin 3539 ExId cexid 32813 SemiGrpcsem 32829 MndOpcmndo 32835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-mndo 32836 |
This theorem is referenced by: mndomgmid 32840 rngo1cl 32908 |
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