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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 12624 | . . 3 ⊢ 0 ∈ ℤ | |
| 2 | 3anass 1095 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴) ↔ (0 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 0 < 𝐴))) | |
| 3 | 1, 2 | mpbiran 709 | . 2 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴) ↔ (𝐴 ∈ ℤ ∧ 0 < 𝐴)) |
| 4 | fzolb 13705 | . 2 ⊢ (0 ∈ (0..^𝐴) ↔ (0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴)) | |
| 5 | elnnz 12623 | . 2 ⊢ (𝐴 ∈ ℕ ↔ (𝐴 ∈ ℤ ∧ 0 < 𝐴)) | |
| 6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ (0 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 0cc0 11155 < clt 11295 ℕcn 12266 ℤcz 12613 ..^cfzo 13694 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 |
| This theorem is referenced by: elfzo0 13740 fzo0n0 13755 fzo0end 13797 fvf1tp 13829 tpf1ofv0 14535 tpfo 14539 wrdsymb1 14591 ccatfv0 14621 ccat1st1st 14666 ccat2s1p1 14667 lswccats1fst 14673 swrdfv0 14687 pfxn0 14724 pfxfv0 14730 pfxtrcfv0 14732 pfx1 14741 cats1un 14759 revs1 14803 repswfsts 14819 cshwidx0mod 14843 cshw1 14860 scshwfzeqfzo 14865 cats1fvn 14897 pfx2 14986 nnnn0modprm0 16844 cshwrepswhash1 17140 efgsval2 19751 efgs1b 19754 efgsp1 19755 efgsres 19756 efgredlemd 19762 efgredlem 19765 efgrelexlemb 19768 pgpfaclem1 20101 dchrisumlem3 27535 tgcgr4 28539 wlkonl1iedg 29683 usgr2pthlem 29783 pthdlem2lem 29787 lfgrn1cycl 29825 uspgrn2crct 29828 crctcshwlkn0lem6 29835 0enwwlksnge1 29884 wwlksm1edg 29901 wwlksnwwlksnon 29935 clwlkclwwlklem2 30019 clwlkclwwlkf1lem3 30025 clwwlkel 30065 clwwlkf1 30068 umgr2cwwk2dif 30083 clwwlknonwwlknonb 30125 upgr3v3e3cycl 30199 upgr4cycl4dv4e 30204 2clwwlk2clwwlk 30369 cycpmco2lem4 33149 cycpmco2lem5 33150 cycpmrn 33163 lmatcl 33815 fib0 34401 signsvtn0 34585 reprpmtf1o 34641 poimirlem3 37630 amgm2d 44211 amgm3d 44212 amgm4d 44213 iccpartigtl 47410 iccpartlt 47411 gpg3kgrtriex 48045 amgmw2d 49323 |
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