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Mirrors > Home > MPE Home > Th. List > dfn2 | Structured version Visualization version GIF version |
Description: The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.) |
Ref | Expression |
---|---|
dfn2 | ⊢ ℕ = (ℕ0 ∖ {0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-n0 12471 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | 1 | difeq1i 4111 | . 2 ⊢ (ℕ0 ∖ {0}) = ((ℕ ∪ {0}) ∖ {0}) |
3 | difun2 4473 | . 2 ⊢ ((ℕ ∪ {0}) ∖ {0}) = (ℕ ∖ {0}) | |
4 | 0nnn 12246 | . . 3 ⊢ ¬ 0 ∈ ℕ | |
5 | difsn 4794 | . . 3 ⊢ (¬ 0 ∈ ℕ → (ℕ ∖ {0}) = ℕ) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (ℕ ∖ {0}) = ℕ |
7 | 2, 3, 6 | 3eqtrri 2757 | 1 ⊢ ℕ = (ℕ0 ∖ {0}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 ∖ cdif 3938 ∪ cun 3939 {csn 4621 0cc0 11107 ℕcn 12210 ℕ0cn0 12470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-nn 12211 df-n0 12471 |
This theorem is referenced by: elnnne0 12484 fcdmnn0supp 12526 fcdmnn0fsupp 12527 fcdmnn0suppg 12528 facnn 14233 fac0 14234 ruclem4 16176 fzo0dvdseq 16265 domnchr 21393 mhpmulcl 22002 logexprlim 27077 eulerpartgbij 33863 eulerpartlemmf 33866 eulerpartlemgf 33870 dffltz 41890 |
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