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| Mirrors > Home > MPE Home > Th. List > Mathboxes > eluzelz2 | Structured version Visualization version GIF version | ||
| Description: A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.) | 
| Ref | Expression | 
|---|---|
| eluzelz2.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) | 
| Ref | Expression | 
|---|---|
| eluzelz2 | ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eluzelz2.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 2 | 1 | eleq2i 2833 | . . 3 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝑀)) | 
| 3 | 2 | biimpi 216 | . 2 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑀)) | 
| 4 | eluzelz 12888 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
| 5 | 3, 4 | syl 17 | 1 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 ℤcz 12613 ℤ≥cuz 12878 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-cnex 11211 ax-resscn 11212 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-neg 11495 df-z 12614 df-uz 12879 | 
| This theorem is referenced by: eluzelz2d 45424 uzublem 45441 uzinico 45573 limsupubuzlem 45727 limsupmnfuzlem 45741 limsupre3uzlem 45750 limsupvaluz2 45753 supcnvlimsup 45755 xlimclim2lem 45854 climxlim2 45861 xlimliminflimsup 45877 smflimmpt 46825 smflimsuplem3 46837 smflimsuplem4 46838 smflimsuplem5 46839 smflimsuplem6 46840 smflimsuplem7 46841 smflimsuplem8 46842 smflimsupmpt 46844 smfliminflem 46845 smfliminfmpt 46847 | 
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