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Mirrors > Home > MPE Home > Th. List > nnnlt1 | Structured version Visualization version GIF version |
Description: A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnnlt1 | ⊢ (𝐴 ∈ ℕ → ¬ 𝐴 < 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1 12271 | . 2 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | |
2 | 1re 11245 | . . 3 ⊢ 1 ∈ ℝ | |
3 | nnre 12250 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
4 | lenlt 11323 | . . 3 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (1 ≤ 𝐴 ↔ ¬ 𝐴 < 1)) | |
5 | 2, 3, 4 | sylancr 586 | . 2 ⊢ (𝐴 ∈ ℕ → (1 ≤ 𝐴 ↔ ¬ 𝐴 < 1)) |
6 | 1, 5 | mpbid 231 | 1 ⊢ (𝐴 ∈ ℕ → ¬ 𝐴 < 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∈ wcel 2099 class class class wbr 5148 ℝcr 11138 1c1 11140 < clt 11279 ≤ cle 11280 ℕcn 12243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 |
This theorem is referenced by: 0nnnALT 12280 nnsub 12287 nn0lt10b 12655 indstr2 12942 sqrt2irr 16226 ovolicc2lem3 25461 nn0prpwlem 35806 |
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