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Mirrors > Home > ILE Home > Th. List > elnnnn0 | GIF version |
Description: The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.) |
Ref | Expression |
---|---|
elnnnn0 | ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nncn 8930 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
2 | npcan1 8338 | . . . . 5 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) | |
3 | 2 | eleq1d 2246 | . . . 4 ⊢ (𝑁 ∈ ℂ → (((𝑁 − 1) + 1) ∈ ℕ ↔ 𝑁 ∈ ℕ)) |
4 | peano2cnm 8226 | . . . . 5 ⊢ (𝑁 ∈ ℂ → (𝑁 − 1) ∈ ℂ) | |
5 | 4 | biantrurd 305 | . . . 4 ⊢ (𝑁 ∈ ℂ → (((𝑁 − 1) + 1) ∈ ℕ ↔ ((𝑁 − 1) ∈ ℂ ∧ ((𝑁 − 1) + 1) ∈ ℕ))) |
6 | 3, 5 | bitr3d 190 | . . 3 ⊢ (𝑁 ∈ ℂ → (𝑁 ∈ ℕ ↔ ((𝑁 − 1) ∈ ℂ ∧ ((𝑁 − 1) + 1) ∈ ℕ))) |
7 | elnn0nn 9221 | . . 3 ⊢ ((𝑁 − 1) ∈ ℕ0 ↔ ((𝑁 − 1) ∈ ℂ ∧ ((𝑁 − 1) + 1) ∈ ℕ)) | |
8 | 6, 7 | bitr4di 198 | . 2 ⊢ (𝑁 ∈ ℂ → (𝑁 ∈ ℕ ↔ (𝑁 − 1) ∈ ℕ0)) |
9 | 1, 8 | biadan2 456 | 1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ0)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∈ wcel 2148 (class class class)co 5878 ℂcc 7812 1c1 7815 + caddc 7817 − cmin 8131 ℕcn 8922 ℕ0cn0 9179 |
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-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 |
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-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-sub 8133 df-inn 8923 df-n0 9180 |
This theorem is referenced by: elfzom1elp1fzo 10205 facnn2 10717 |
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