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Mirrors > Home > MPE Home > Th. List > elnnne0 | Structured version Visualization version GIF version |
Description: The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Ref | Expression |
---|---|
elnnne0 | ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfn2 12531 | . . 3 ⊢ ℕ = (ℕ0 ∖ {0}) | |
2 | 1 | eleq2i 2818 | . 2 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℕ0 ∖ {0})) |
3 | eldifsn 4785 | . 2 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) | |
4 | 2, 3 | bitri 274 | 1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3943 {csn 4623 0cc0 11149 ℕcn 12258 ℕ0cn0 12518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-nn 12259 df-n0 12519 |
This theorem is referenced by: nn0n0n1ge2 12585 nn0nndivcl 12589 fzo1fzo0n0 13731 elfznelfzo 13786 hashnn0n0nn 14403 swrdccatin1 14728 cshwsublen 14799 cshwidxmod 14806 cshwidx0 14809 repswcshw 14815 cshw1 14825 nn0onn 16377 hashfinmndnn 18739 odhash3 19570 prmgrpsimpgd 20110 0ringnnzr 20503 psdmul 22156 cply1mul 22284 fvmptnn04if 22839 chfacfisf 22844 chfacfisfcpmat 22845 tayl0 26386 dvtaylp 26395 2sqmod 27462 wlkonl1iedg 29599 pthdlem2 29702 crctcsh 29755 clwwlkneq0 29959 hashecclwwlkn1 30007 umgrhashecclwwlk 30008 clwwlknon0 30023 frgrreg 30324 frgrregord013 30325 xnn0gt0 32676 subne0nn 32725 chnind 32883 chnub 32884 plymulx0 34406 plymulx 34407 signstfvn 34428 signstfveq0a 34435 poimirlem13 37347 poimirlem20 37354 aks6d1c4 41836 aks6d1c7lem1 41892 flt0 42327 dvnmul 45600 dvnprodlem3 45605 wallispilem3 45724 fourierdlem103 45866 fourierdlem104 45867 etransclem28 45919 etransclem35 45926 etransclem38 45929 etransclem44 45935 2ffzoeq 46976 lswn0 47052 ztprmneprm 47762 |
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