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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 12068 | . . 3 ⊢ ℕ = (ℕ0 ∖ {0}) | |
2 | 1 | eleq2i 2822 | . 2 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℕ0 ∖ {0})) |
3 | eldifsn 4686 | . 2 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) | |
4 | 2, 3 | bitri 278 | 1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2112 ≠ wne 2932 ∖ cdif 3850 {csn 4527 0cc0 10694 ℕcn 11795 ℕ0cn0 12055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-nn 11796 df-n0 12056 |
This theorem is referenced by: nn0n0n1ge2 12122 nn0nndivcl 12126 fzo1fzo0n0 13258 elfznelfzo 13312 hashnn0n0nn 13923 swrdccatin1 14255 cshwsublen 14326 cshwidxmod 14333 cshwidx0 14336 repswcshw 14342 cshw1 14352 nn0onn 15904 hashfinmndnn 18144 odhash3 18919 prmgrpsimpgd 19455 0ringnnzr 20261 cply1mul 21169 fvmptnn04if 21700 chfacfisf 21705 chfacfisfcpmat 21706 tayl0 25208 dvtaylp 25216 2sqmod 26271 wlkonl1iedg 27707 pthdlem2 27809 crctcsh 27862 clwwlkneq0 28066 hashecclwwlkn1 28114 umgrhashecclwwlk 28115 clwwlknon0 28130 frgrreg 28431 frgrregord013 28432 xnn0gt0 30766 subne0nn 30809 plymulx0 32192 plymulx 32193 signstfvn 32214 signstfveq0a 32221 poimirlem13 35476 poimirlem20 35483 flt0 40118 dvnmul 43102 dvnprodlem3 43107 wallispilem3 43226 fourierdlem103 43368 fourierdlem104 43369 etransclem28 43421 etransclem35 43428 etransclem38 43431 etransclem44 43437 2ffzoeq 44436 lswn0 44512 ztprmneprm 45299 |
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