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Mirrors > Home > MPE Home > Th. List > elnnnn0b | Structured version Visualization version GIF version |
Description: The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
Ref | Expression |
---|---|
elnnnn0b | ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 11905 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | nngt0 11669 | . . 3 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
3 | 1, 2 | jca 514 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) |
4 | elnn0 11900 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
5 | breq2 5070 | . . . . . 6 ⊢ (𝑁 = 0 → (0 < 𝑁 ↔ 0 < 0)) | |
6 | 0re 10643 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
7 | 6 | ltnri 10749 | . . . . . . 7 ⊢ ¬ 0 < 0 |
8 | 7 | pm2.21i 119 | . . . . . 6 ⊢ (0 < 0 → 𝑁 ∈ ℕ) |
9 | 5, 8 | syl6bi 255 | . . . . 5 ⊢ (𝑁 = 0 → (0 < 𝑁 → 𝑁 ∈ ℕ)) |
10 | 9 | jao1i 854 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (0 < 𝑁 → 𝑁 ∈ ℕ)) |
11 | 4, 10 | sylbi 219 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0 < 𝑁 → 𝑁 ∈ ℕ)) |
12 | 11 | imp 409 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 0 < 𝑁) → 𝑁 ∈ ℕ) |
13 | 3, 12 | impbii 211 | 1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 0cc0 10537 < clt 10675 ℕcn 11638 ℕ0cn0 11898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 |
This theorem is referenced by: elnnnn0c 11943 nn0p1elfzo 13081 bccl2 13684 ccatfv0 13937 ccat2s1fvw 13998 ccat2s1fvwOLD 13999 swrdswrd 14067 cycpmfv2 30756 eulerpartlems 31618 rmxnn 39568 fsumnncl 41872 ioodvbdlimc1lem2 42237 ioodvbdlimc2lem 42239 |
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