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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 12504 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | 1 | difeq1i 4116 | . 2 ⊢ (ℕ0 ∖ {0}) = ((ℕ ∪ {0}) ∖ {0}) |
3 | difun2 4481 | . 2 ⊢ ((ℕ ∪ {0}) ∖ {0}) = (ℕ ∖ {0}) | |
4 | 0nnn 12279 | . . 3 ⊢ ¬ 0 ∈ ℕ | |
5 | difsn 4802 | . . 3 ⊢ (¬ 0 ∈ ℕ → (ℕ ∖ {0}) = ℕ) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (ℕ ∖ {0}) = ℕ |
7 | 2, 3, 6 | 3eqtrri 2761 | 1 ⊢ ℕ = (ℕ0 ∖ {0}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 ∖ cdif 3944 ∪ cun 3945 {csn 4629 0cc0 11139 ℕcn 12243 ℕ0cn0 12503 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-nn 12244 df-n0 12504 |
This theorem is referenced by: elnnne0 12517 fcdmnn0supp 12559 fcdmnn0fsupp 12560 fcdmnn0suppg 12561 facnn 14267 fac0 14268 ruclem4 16211 fzo0dvdseq 16300 domnchr 21462 mhpmulcl 22073 logexprlim 27171 eulerpartgbij 33992 eulerpartlemmf 33995 eulerpartlemgf 33999 dffltz 42058 |
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