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Mirrors > Home > MPE Home > Th. List > nn0lt10b | Structured version Visualization version GIF version |
Description: A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by OpenAI, 25-Mar-2020.) |
Ref | Expression |
---|---|
nn0lt10b | ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 11887 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | nnnlt1 11657 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 < 1) | |
3 | 2 | pm2.21d 121 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 < 1 → 𝑁 = 0)) |
4 | ax-1 6 | . . . 4 ⊢ (𝑁 = 0 → (𝑁 < 1 → 𝑁 = 0)) | |
5 | 3, 4 | jaoi 851 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑁 < 1 → 𝑁 = 0)) |
6 | 1, 5 | sylbi 218 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 → 𝑁 = 0)) |
7 | 0lt1 11150 | . . 3 ⊢ 0 < 1 | |
8 | breq1 5060 | . . 3 ⊢ (𝑁 = 0 → (𝑁 < 1 ↔ 0 < 1)) | |
9 | 7, 8 | mpbiri 259 | . 2 ⊢ (𝑁 = 0 → 𝑁 < 1) |
10 | 6, 9 | impbid1 226 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∨ wo 841 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 0cc0 10525 1c1 10526 < clt 10663 ℕcn 11626 ℕ0cn0 11885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 |
This theorem is referenced by: nn0lt2 12033 nn0le2is012 12034 fz1n 12913 zdis 23351 plyrem 24821 efif1olem4 25056 acycgr1v 32293 poimirlem28 34801 |
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