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Mirrors > Home > MPE Home > Th. List > elfzuz2 | Structured version Visualization version GIF version |
Description: Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfzuz2 | ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuzb 12586 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) | |
2 | eqid 2797 | . . 3 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
3 | 2 | uztrn2 11944 | . 2 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → 𝑁 ∈ (ℤ≥‘𝑀)) |
4 | 1, 3 | sylbi 209 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ‘cfv 6099 (class class class)co 6876 ℤ≥cuz 11926 ...cfz 12576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-pre-lttri 10296 ax-pre-lttrn 10297 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-1st 7399 df-2nd 7400 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-neg 10557 df-z 11663 df-uz 11927 df-fz 12577 |
This theorem is referenced by: elfzle3 12597 elfzubelfz 12603 fzn0 12605 fzopth 12628 elfzmlbm 12700 elfzom1elp1fzo 12786 elfzr 12832 elfzlmr 12833 bcm1k 13351 bcpasc 13357 seqcoll 13493 swrdccatin12lem2c 13788 pfxccatin12 13792 swrdccatin12OLD 13793 splid 13825 splidOLD 13826 spllen 13827 spllenOLD 13828 prmodvdslcmf 16081 gexcl3 18312 dvn2bss 24031 pserdvlem2 24520 ppinprm 25227 chtnprm 25229 chpval2 25292 chpchtsum 25293 lgsdir2lem2 25400 fzto1stfv1 30359 fzto1stinvn 30362 wrdsplex 31127 wrdsplexOLD 31128 monoords 40244 |
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