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Mirrors > Home > ILE Home > Th. List > dfn2 | 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 9153 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | 1 | difeq1i 3249 | . 2 ⊢ (ℕ0 ∖ {0}) = ((ℕ ∪ {0}) ∖ {0}) |
3 | difun2 3502 | . 2 ⊢ ((ℕ ∪ {0}) ∖ {0}) = (ℕ ∖ {0}) | |
4 | 0nnn 8922 | . . 3 ⊢ ¬ 0 ∈ ℕ | |
5 | difsn 3728 | . . 3 ⊢ (¬ 0 ∈ ℕ → (ℕ ∖ {0}) = ℕ) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (ℕ ∖ {0}) = ℕ |
7 | 2, 3, 6 | 3eqtrri 2203 | 1 ⊢ ℕ = (ℕ0 ∖ {0}) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1353 ∈ wcel 2148 ∖ cdif 3126 ∪ cun 3127 {csn 3591 0cc0 7789 ℕcn 8895 ℕ0cn0 9152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-pre-ltirr 7901 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-xp 4628 df-cnv 4630 df-iota 5173 df-fv 5219 df-ov 5871 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-inn 8896 df-n0 9153 |
This theorem is referenced by: elnnne0 9166 nn0supp 9204 facnn 10678 fac0 10679 fzo0dvdseq 11833 |
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