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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 11265 | . . 3 ⊢ 0 ∈ ℤ | |
2 | 3anass 1035 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴) ↔ (0 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 0 < 𝐴))) | |
3 | 1, 2 | mpbiran 955 | . 2 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴) ↔ (𝐴 ∈ ℤ ∧ 0 < 𝐴)) |
4 | fzolb 12345 | . 2 ⊢ (0 ∈ (0..^𝐴) ↔ (0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴)) | |
5 | elnnz 11264 | . 2 ⊢ (𝐴 ∈ ℕ ↔ (𝐴 ∈ ℤ ∧ 0 < 𝐴)) | |
6 | 3, 4, 5 | 3bitr4i 291 | 1 ⊢ (0 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 0cc0 9815 < clt 9953 ℕcn 10897 ℤcz 11254 ..^cfzo 12334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 |
This theorem is referenced by: elfzo0 12376 fzo0n0 12387 fzo0end 12426 wrdsymb1 13197 ccatfv0 13220 ccat2s1p1 13257 lswccats1fst 13264 swrdn0 13282 swrd0fv0 13292 swrdtrcfv0 13294 cats1un 13327 revs1 13365 repswfsts 13379 cshwidx0mod 13402 cshw1 13419 scshwfzeqfzo 13423 cats1fvn 13454 nnnn0modprm0 15349 cshwrepswhash1 15647 efgsval2 17969 efgs1b 17972 efgsp1 17973 efgsres 17974 efgredlemd 17980 efgredlem 17983 efgrelexlemb 17986 pgpfaclem1 18303 dchrisumlem3 24980 tgcgr4 25226 usgrcyclnl2 26169 4cycl4dv 26195 wwlkm1edg 26263 clwlkisclwwlklem1 26315 clwwlkel 26321 clwwlkf1 26324 usg2cwwk2dif 26348 clwlkf1clwwlklem 26376 lmatcl 29210 fib0 29788 signsvtn0 29973 poimirlem3 32582 amgm2d 37523 amgm3d 37524 amgm4d 37525 iccpartigtl 39961 iccpartlt 39962 pfxfv0 40263 pfxtrcfv0 40265 pfx1 40274 pfx2 40275 wlkOnl1iedg 40873 usgr2pthlem 40969 pthdlem2lem 40973 lfgrn1cycl 41008 uspgrn2crct 41011 crctcsh1wlkn0lem6 41018 0enwwlksnge1 41060 wwlksm1edg 41078 wwlksnwwlksnon 41121 clwlkclwwlklem2 41209 clwwlksel 41221 clwwlksf1 41224 umgr2cwwk2dif 41248 clwlksf1clwwlklem 41275 upgr3v3e3cycl 41347 upgr4cycl4dv4e 41352 amgmw2d 42359 |
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