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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 8737 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | 1 | difeq1i 3117 | . 2 ⊢ (ℕ0 ∖ {0}) = ((ℕ ∪ {0}) ∖ {0}) |
3 | difun2 3368 | . 2 ⊢ ((ℕ ∪ {0}) ∖ {0}) = (ℕ ∖ {0}) | |
4 | 0nnn 8512 | . . 3 ⊢ ¬ 0 ∈ ℕ | |
5 | difsn 3582 | . . 3 ⊢ (¬ 0 ∈ ℕ → (ℕ ∖ {0}) = ℕ) | |
6 | 4, 5 | ax-mp 7 | . 2 ⊢ (ℕ ∖ {0}) = ℕ |
7 | 2, 3, 6 | 3eqtrri 2114 | 1 ⊢ ℕ = (ℕ0 ∖ {0}) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1290 ∈ wcel 1439 ∖ cdif 2999 ∪ cun 3000 {csn 3452 0cc0 7413 ℕcn 8485 ℕ0cn0 8736 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-cnex 7499 ax-resscn 7500 ax-1re 7502 ax-addrcl 7505 ax-0lt1 7514 ax-0id 7516 ax-rnegex 7517 ax-pre-ltirr 7520 ax-pre-lttrn 7522 ax-pre-ltadd 7524 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2624 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-br 3854 df-opab 3908 df-xp 4460 df-cnv 4462 df-iota 4995 df-fv 5038 df-ov 5671 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-inn 8486 df-n0 8737 |
This theorem is referenced by: elnnne0 8750 nn0supp 8788 facnn 10198 fac0 10199 fzo0dvdseq 11199 |
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