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Mirrors > Home > MPE Home > Th. List > lbfzo0 | Structured version Visualization version GIF version |
Description: An integer is strictly greater than zero iff it is a member of ℕ. (Contributed by Mario Carneiro, 29-Sep-2015.) |
Ref | Expression |
---|---|
lbfzo0 | ⊢ (0 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 11980 | . . 3 ⊢ 0 ∈ ℤ | |
2 | 3anass 1087 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴) ↔ (0 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 0 < 𝐴))) | |
3 | 1, 2 | mpbiran 705 | . 2 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴) ↔ (𝐴 ∈ ℤ ∧ 0 < 𝐴)) |
4 | fzolb 13032 | . 2 ⊢ (0 ∈ (0..^𝐴) ↔ (0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴)) | |
5 | elnnz 11979 | . 2 ⊢ (𝐴 ∈ ℕ ↔ (𝐴 ∈ ℤ ∧ 0 < 𝐴)) | |
6 | 3, 4, 5 | 3bitr4i 304 | 1 ⊢ (0 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 0cc0 10525 < clt 10663 ℕcn 11626 ℤcz 11969 ..^cfzo 13021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 |
This theorem is referenced by: elfzo0 13066 fzo0n0 13077 fzo0end 13117 wrdsymb1 13893 ccatfv0 13925 ccat1st1st 13972 ccat2s1p1 13973 ccat2s1p1OLD 13975 lswccats1fst 13982 swrdfv0 13999 pfxn0 14036 pfxfv0 14042 pfxtrcfv0 14044 pfx1 14053 cats1un 14071 revs1 14115 repswfsts 14131 cshwidx0mod 14155 cshw1 14172 scshwfzeqfzo 14176 cats1fvn 14208 pfx2 14297 nnnn0modprm0 16131 cshwrepswhash1 16424 efgsval2 18788 efgs1b 18791 efgsp1 18792 efgsres 18793 efgredlemd 18799 efgredlem 18802 efgrelexlemb 18805 pgpfaclem1 19132 dchrisumlem3 25994 tgcgr4 26244 wlkonl1iedg 27374 usgr2pthlem 27471 pthdlem2lem 27475 lfgrn1cycl 27510 uspgrn2crct 27513 crctcshwlkn0lem6 27520 0enwwlksnge1 27569 wwlksm1edg 27586 wwlksnwwlksnon 27621 clwlkclwwlklem2 27705 clwlkclwwlkf1lem3 27711 clwwlkel 27752 clwwlkf1 27755 umgr2cwwk2dif 27770 clwwlknonwwlknonb 27812 upgr3v3e3cycl 27886 upgr4cycl4dv4e 27891 2clwwlk2clwwlk 28056 cycpmco2lem4 30698 cycpmco2lem5 30699 cycpmrn 30712 lmatcl 30980 fib0 31556 signsvtn0 31739 reprpmtf1o 31796 poimirlem3 34776 amgm2d 40429 amgm3d 40430 amgm4d 40431 iccpartigtl 43460 iccpartlt 43461 amgmw2d 44833 |
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