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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 12833 | . 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 2106 ‘cfv 6540 ℤcz 12554 ℤ≥cuz 12818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-neg 11443 df-z 12555 df-uz 12819 |
This theorem is referenced by: ccatass 14534 ccatrn 14535 swrdccat2 14615 pfxccat1 14648 splfv1 14701 splval2 14703 revccat 14712 ntrivcvgn0 15840 gsumsplit1r 18602 gsumsgrpccat 18717 efginvrel2 19589 signstfvp 33570 poimirlem20 36496 aks4d1p1p3 40922 aks4d1p1p4 40924 aks4d1p1p6 40926 aks4d1p1p7 40927 aks4d1p1p5 40928 aks4d1p1 40929 aks4d1p6 40934 sumcubes 41206 uzidd2 44112 uzinico3 44262 smflimsuplem7 45528 smflimsuplem8 45529 smflimsupmpt 45531 smfliminfmpt 45534 |
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