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| Mirrors > Home > MPE Home > Th. List > uzidd | Structured version Visualization version GIF version | ||
| Description: Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) | 
| Ref | Expression | 
|---|---|
| uzidd.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| Ref | Expression | 
|---|---|
| uzidd | ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uzidd.1 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 2 | uzid 12894 | . 2 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ‘cfv 6560 ℤcz 12615 ℤ≥cuz 12879 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-neg 11496 df-z 12616 df-uz 12880 | 
| This theorem is referenced by: fzdif1 13646 ccatass 14627 ccatrn 14628 swrdccat2 14708 pfxccat1 14741 splfv1 14794 splval2 14796 revccat 14805 ntrivcvgn0 15935 gsumsplit1r 18701 gsumsgrpccat 18854 efginvrel2 19746 signstfvp 34587 poimirlem20 37648 aks4d1p1p3 42071 aks4d1p1p4 42073 aks4d1p1p6 42075 aks4d1p1p7 42076 aks4d1p1p5 42077 aks4d1p1 42078 aks4d1p6 42083 sumcubes 42352 uzidd2 45432 uzinico3 45581 smflimsuplem7 46846 smflimsuplem8 46847 smflimsupmpt 46849 smfliminfmpt 46852 | 
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