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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 11725 | . . 3 ⊢ ℕ = (ℕ0 ∖ {0}) | |
2 | 1 | eleq2i 2857 | . 2 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℕ0 ∖ {0})) |
3 | eldifsn 4594 | . 2 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) | |
4 | 2, 3 | bitri 267 | 1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 ∈ wcel 2050 ≠ wne 2967 ∖ cdif 3828 {csn 4442 0cc0 10337 ℕcn 11441 ℕ0cn0 11710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-ov 6981 df-om 7399 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-nn 11442 df-n0 11711 |
This theorem is referenced by: nn0n0n1ge2 11777 nn0nndivcl 11781 fzo1fzo0n0 12906 elfznelfzo 12960 hashnn0n0nn 13568 ccat1st1st 13794 swrdccatin1 13927 cshwsublen 14023 cshwidxmod 14030 cshwidx0 14033 repswcshw 14039 cshw1 14049 nn0onn 15594 odhash3 18465 0ringnnzr 19766 cply1mul 20168 fvmptnn04if 21164 chfacfisf 21169 chfacfisfcpmat 21170 tayl0 24656 dvtaylp 24664 2sqmod 25717 wlkonl1iedg 27152 pthdlem2 27260 crctcsh 27313 clwwlkneq0 27547 hashecclwwlkn1 27604 umgrhashecclwwlk 27605 clwwlknon0 27624 frgrreg 27954 frgrregord013 27955 xnn0gt0 30249 plymulx0 31463 plymulx 31464 signstfvn 31485 signstfveq0a 31493 poimirlem13 34346 poimirlem20 34353 hashfinmndnn 40022 prmgrpsimpgd 40049 dvnmul 41659 dvnprodlem3 41664 wallispilem3 41784 fourierdlem103 41926 fourierdlem104 41927 etransclem28 41979 etransclem35 41986 etransclem38 41989 etransclem44 41995 2ffzoeq 42935 lswn0 42977 ztprmneprm 43760 |
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