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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 12629 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) | |
2 | eqid 2825 | . . 3 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
3 | 2 | uztrn2 11986 | . 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 386 ∈ wcel 2164 ‘cfv 6123 (class class class)co 6905 ℤ≥cuz 11968 ...cfz 12619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-pre-lttri 10326 ax-pre-lttrn 10327 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-neg 10588 df-z 11705 df-uz 11969 df-fz 12620 |
This theorem is referenced by: elfzle3 12640 elfzubelfz 12646 fzn0 12648 fzopth 12671 elfzmlbm 12744 elfzom1elp1fzo 12830 elfzr 12876 elfzlmr 12877 bcm1k 13395 bcpasc 13401 seqcoll 13537 swrdccatin12lem2c 13827 pfxccatin12 13831 swrdccatin12OLD 13832 splid 13864 splidOLD 13865 spllen 13866 spllenOLD 13867 prmodvdslcmf 16122 gexcl3 18353 dvn2bss 24092 pserdvlem2 24581 ppinprm 25291 chtnprm 25293 chpval2 25356 chpchtsum 25357 lgsdir2lem2 25464 fzto1stfv1 30385 fzto1stinvn 30388 wrdsplex 31153 wrdsplexOLD 31154 monoords 40302 |
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