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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 11957 | . 2 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) | |
2 | nn0n0n1ge2b 11951 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) | |
3 | 0nn0 11900 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
4 | nn0nepnf 11963 | . . . . . . . 8 ⊢ (0 ∈ ℕ0 → 0 ≠ +∞) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 0 ≠ +∞ |
6 | 5 | necomi 3041 | . . . . . 6 ⊢ +∞ ≠ 0 |
7 | neeq1 3049 | . . . . . 6 ⊢ (𝑁 = +∞ → (𝑁 ≠ 0 ↔ +∞ ≠ 0)) | |
8 | 6, 7 | mpbiri 261 | . . . . 5 ⊢ (𝑁 = +∞ → 𝑁 ≠ 0) |
9 | 1nn0 11901 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
10 | nn0nepnf 11963 | . . . . . . . 8 ⊢ (1 ∈ ℕ0 → 1 ≠ +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ 1 ≠ +∞ |
12 | 11 | necomi 3041 | . . . . . 6 ⊢ +∞ ≠ 1 |
13 | neeq1 3049 | . . . . . 6 ⊢ (𝑁 = +∞ → (𝑁 ≠ 1 ↔ +∞ ≠ 1)) | |
14 | 12, 13 | mpbiri 261 | . . . . 5 ⊢ (𝑁 = +∞ → 𝑁 ≠ 1) |
15 | 8, 14 | jca 515 | . . . 4 ⊢ (𝑁 = +∞ → (𝑁 ≠ 0 ∧ 𝑁 ≠ 1)) |
16 | 2re 11699 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
17 | 16 | rexri 10688 | . . . . . 6 ⊢ 2 ∈ ℝ* |
18 | pnfge 12513 | . . . . . 6 ⊢ (2 ∈ ℝ* → 2 ≤ +∞) | |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ 2 ≤ +∞ |
20 | breq2 5034 | . . . . 5 ⊢ (𝑁 = +∞ → (2 ≤ 𝑁 ↔ 2 ≤ +∞)) | |
21 | 19, 20 | mpbiri 261 | . . . 4 ⊢ (𝑁 = +∞ → 2 ≤ 𝑁) |
22 | 15, 21 | 2thd 268 | . . 3 ⊢ (𝑁 = +∞ → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
23 | 2, 22 | jaoi 854 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∨ 𝑁 = +∞) → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
24 | 1, 23 | sylbi 220 | 1 ⊢ (𝑁 ∈ ℕ0* → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 0cc0 10526 1c1 10527 +∞cpnf 10661 ℝ*cxr 10663 ≤ cle 10665 2c2 11680 ℕ0cn0 11885 ℕ0*cxnn0 11955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 |
This theorem is referenced by: vdgfrgrgt2 28083 xnn01gt 30521 lfuhgr2 32478 |
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