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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 11993 | . . 3 ⊢ 0 ∈ ℤ | |
2 | 3anass 1091 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴) ↔ (0 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 0 < 𝐴))) | |
3 | 1, 2 | mpbiran 707 | . 2 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴) ↔ (𝐴 ∈ ℤ ∧ 0 < 𝐴)) |
4 | fzolb 13045 | . 2 ⊢ (0 ∈ (0..^𝐴) ↔ (0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴)) | |
5 | elnnz 11992 | . 2 ⊢ (𝐴 ∈ ℕ ↔ (𝐴 ∈ ℤ ∧ 0 < 𝐴)) | |
6 | 3, 4, 5 | 3bitr4i 305 | 1 ⊢ (0 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 0cc0 10537 < clt 10675 ℕcn 11638 ℤcz 11982 ..^cfzo 13034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 |
This theorem is referenced by: elfzo0 13079 fzo0n0 13090 fzo0end 13130 wrdsymb1 13905 ccatfv0 13937 ccat1st1st 13984 ccat2s1p1 13985 ccat2s1p1OLD 13987 lswccats1fst 13994 swrdfv0 14011 pfxn0 14048 pfxfv0 14054 pfxtrcfv0 14056 pfx1 14065 cats1un 14083 revs1 14127 repswfsts 14143 cshwidx0mod 14167 cshw1 14184 scshwfzeqfzo 14188 cats1fvn 14220 pfx2 14309 nnnn0modprm0 16143 cshwrepswhash1 16436 efgsval2 18859 efgs1b 18862 efgsp1 18863 efgsres 18864 efgredlemd 18870 efgredlem 18873 efgrelexlemb 18876 pgpfaclem1 19203 dchrisumlem3 26067 tgcgr4 26317 wlkonl1iedg 27447 usgr2pthlem 27544 pthdlem2lem 27548 lfgrn1cycl 27583 uspgrn2crct 27586 crctcshwlkn0lem6 27593 0enwwlksnge1 27642 wwlksm1edg 27659 wwlksnwwlksnon 27694 clwlkclwwlklem2 27778 clwlkclwwlkf1lem3 27784 clwwlkel 27825 clwwlkf1 27828 umgr2cwwk2dif 27843 clwwlknonwwlknonb 27885 upgr3v3e3cycl 27959 upgr4cycl4dv4e 27964 2clwwlk2clwwlk 28129 cycpmco2lem4 30771 cycpmco2lem5 30772 cycpmrn 30785 lmatcl 31081 fib0 31657 signsvtn0 31840 reprpmtf1o 31897 poimirlem3 34910 amgm2d 40571 amgm3d 40572 amgm4d 40573 iccpartigtl 43603 iccpartlt 43604 amgmw2d 44925 |
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