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Mirrors > Home > MPE Home > Th. List > xnn0n0n1ge2b | Structured version Visualization version GIF version |
Description: An extended nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by AV, 5-Apr-2021.) |
Ref | Expression |
---|---|
xnn0n0n1ge2b | ⊢ (𝑁 ∈ ℕ0* → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxnn0 11716 | . 2 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) | |
2 | nn0n0n1ge2b 11710 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) | |
3 | 0nn0 11659 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
4 | nn0nepnf 11722 | . . . . . . . 8 ⊢ (0 ∈ ℕ0 → 0 ≠ +∞) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 0 ≠ +∞ |
6 | 5 | necomi 3022 | . . . . . 6 ⊢ +∞ ≠ 0 |
7 | neeq1 3030 | . . . . . 6 ⊢ (𝑁 = +∞ → (𝑁 ≠ 0 ↔ +∞ ≠ 0)) | |
8 | 6, 7 | mpbiri 250 | . . . . 5 ⊢ (𝑁 = +∞ → 𝑁 ≠ 0) |
9 | 1nn0 11660 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
10 | nn0nepnf 11722 | . . . . . . . 8 ⊢ (1 ∈ ℕ0 → 1 ≠ +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ 1 ≠ +∞ |
12 | 11 | necomi 3022 | . . . . . 6 ⊢ +∞ ≠ 1 |
13 | neeq1 3030 | . . . . . 6 ⊢ (𝑁 = +∞ → (𝑁 ≠ 1 ↔ +∞ ≠ 1)) | |
14 | 12, 13 | mpbiri 250 | . . . . 5 ⊢ (𝑁 = +∞ → 𝑁 ≠ 1) |
15 | 8, 14 | jca 507 | . . . 4 ⊢ (𝑁 = +∞ → (𝑁 ≠ 0 ∧ 𝑁 ≠ 1)) |
16 | 2re 11449 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
17 | 16 | rexri 10435 | . . . . . 6 ⊢ 2 ∈ ℝ* |
18 | pnfge 12275 | . . . . . 6 ⊢ (2 ∈ ℝ* → 2 ≤ +∞) | |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ 2 ≤ +∞ |
20 | breq2 4890 | . . . . 5 ⊢ (𝑁 = +∞ → (2 ≤ 𝑁 ↔ 2 ≤ +∞)) | |
21 | 19, 20 | mpbiri 250 | . . . 4 ⊢ (𝑁 = +∞ → 2 ≤ 𝑁) |
22 | 15, 21 | 2thd 257 | . . 3 ⊢ (𝑁 = +∞ → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
23 | 2, 22 | jaoi 846 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∨ 𝑁 = +∞) → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
24 | 1, 23 | sylbi 209 | 1 ⊢ (𝑁 ∈ ℕ0* → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 class class class wbr 4886 0cc0 10272 1c1 10273 +∞cpnf 10408 ℝ*cxr 10410 ≤ cle 10412 2c2 11430 ℕ0cn0 11642 ℕ0*cxnn0 11714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-n0 11643 df-xnn0 11715 |
This theorem is referenced by: vdgfrgrgt2 27720 |
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