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Mirrors > Home > ILE Home > Th. List > elnn0nn | GIF version |
Description: The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
elnn0nn | ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0cn 9105 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
2 | nn0p1nn 9134 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) | |
3 | 1, 2 | jca 304 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ)) |
4 | simpl 108 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → 𝑁 ∈ ℂ) | |
5 | ax-1cn 7827 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | pncan 8085 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁) | |
7 | 4, 5, 6 | sylancl 410 | . . 3 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → ((𝑁 + 1) − 1) = 𝑁) |
8 | nnm1nn0 9136 | . . . 4 ⊢ ((𝑁 + 1) ∈ ℕ → ((𝑁 + 1) − 1) ∈ ℕ0) | |
9 | 8 | adantl 275 | . . 3 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → ((𝑁 + 1) − 1) ∈ ℕ0) |
10 | 7, 9 | eqeltrrd 2235 | . 2 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → 𝑁 ∈ ℕ0) |
11 | 3, 10 | impbii 125 | 1 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1335 ∈ wcel 2128 (class class class)co 5826 ℂcc 7732 1c1 7735 + caddc 7737 − cmin 8050 ℕcn 8838 ℕ0cn0 9095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-addcom 7834 ax-addass 7836 ax-distr 7838 ax-i2m1 7839 ax-0id 7842 ax-rnegex 7843 ax-cnre 7845 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-sub 8052 df-inn 8839 df-n0 9096 |
This theorem is referenced by: elnnnn0 9138 peano2z 9208 |
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