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Mirrors > Home > ILE Home > Th. List > eluzel2 | GIF version |
Description: Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
eluzel2 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzf 9353 | . . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
2 | frel 5285 | . . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → Rel ℤ≥) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ Rel ℤ≥ |
4 | relelfvdm 5461 | . . 3 ⊢ ((Rel ℤ≥ ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ dom ℤ≥) | |
5 | 3, 4 | mpan 421 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ dom ℤ≥) |
6 | 1 | fdmi 5288 | . 2 ⊢ dom ℤ≥ = ℤ |
7 | 5, 6 | eleqtrdi 2233 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 𝒫 cpw 3515 dom cdm 4547 Rel wrel 4552 ⟶wf 5127 ‘cfv 5131 ℤcz 9078 ℤ≥cuz 9350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-neg 7960 df-z 9079 df-uz 9351 |
This theorem is referenced by: eluz2 9356 uztrn 9366 uzneg 9368 uzss 9370 uz11 9372 eluzadd 9378 uzm1 9380 uzin 9382 uzind4 9410 elfz5 9829 elfzel1 9836 eluzfz1 9842 fzsplit2 9861 fzopth 9872 fzpred 9881 fzpreddisj 9882 fzdifsuc 9892 uzsplit 9903 uzdisj 9904 elfzp12 9910 fzm1 9911 uznfz 9914 nn0disj 9946 fzolb 9961 fzoss2 9980 fzouzdisj 9988 ige2m2fzo 10006 elfzonelfzo 10038 frec2uzrand 10209 frecfzen2 10231 seq3p1 10266 seqp1cd 10270 seq3clss 10271 seq3feq2 10274 seq3fveq 10275 seq3shft2 10277 ser3mono 10282 seq3split 10283 seq3caopr3 10285 seq3caopr2 10286 seq3f1olemp 10306 seq3f1oleml 10307 seq3f1o 10308 seq3id3 10311 seq3id 10312 seq3homo 10314 seq3z 10315 seq3distr 10317 ser3ge0 10321 ser3le 10322 leexp2a 10377 hashfz 10599 hashfzo 10600 hashfzp1 10602 seq3coll 10617 rexanuz2 10795 cau4 10920 clim2ser 11138 clim2ser2 11139 climserle 11146 fsum3cvg 11179 fsum3cvg2 11195 fsumsersdc 11196 fsum3ser 11198 fsumm1 11217 fsum1p 11219 telfsumo 11267 fsumparts 11271 cvgcmpub 11277 isumsplit 11292 cvgratnnlemmn 11326 clim2prod 11340 clim2divap 11341 prodfrecap 11347 prodfdivap 11348 ntrivcvgap 11349 fproddccvg 11373 inffz 13429 |
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