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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 12598 | . 2 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) | |
2 | nn0n0n1ge2b 12592 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) | |
3 | 0nn0 12539 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
4 | nn0nepnf 12604 | . . . . . . . 8 ⊢ (0 ∈ ℕ0 → 0 ≠ +∞) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 0 ≠ +∞ |
6 | 5 | necomi 2985 | . . . . . 6 ⊢ +∞ ≠ 0 |
7 | neeq1 2993 | . . . . . 6 ⊢ (𝑁 = +∞ → (𝑁 ≠ 0 ↔ +∞ ≠ 0)) | |
8 | 6, 7 | mpbiri 257 | . . . . 5 ⊢ (𝑁 = +∞ → 𝑁 ≠ 0) |
9 | 1nn0 12540 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
10 | nn0nepnf 12604 | . . . . . . . 8 ⊢ (1 ∈ ℕ0 → 1 ≠ +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ 1 ≠ +∞ |
12 | 11 | necomi 2985 | . . . . . 6 ⊢ +∞ ≠ 1 |
13 | neeq1 2993 | . . . . . 6 ⊢ (𝑁 = +∞ → (𝑁 ≠ 1 ↔ +∞ ≠ 1)) | |
14 | 12, 13 | mpbiri 257 | . . . . 5 ⊢ (𝑁 = +∞ → 𝑁 ≠ 1) |
15 | 8, 14 | jca 510 | . . . 4 ⊢ (𝑁 = +∞ → (𝑁 ≠ 0 ∧ 𝑁 ≠ 1)) |
16 | 2re 12338 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
17 | 16 | rexri 11322 | . . . . . 6 ⊢ 2 ∈ ℝ* |
18 | pnfge 13164 | . . . . . 6 ⊢ (2 ∈ ℝ* → 2 ≤ +∞) | |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ 2 ≤ +∞ |
20 | breq2 5157 | . . . . 5 ⊢ (𝑁 = +∞ → (2 ≤ 𝑁 ↔ 2 ≤ +∞)) | |
21 | 19, 20 | mpbiri 257 | . . . 4 ⊢ (𝑁 = +∞ → 2 ≤ 𝑁) |
22 | 15, 21 | 2thd 264 | . . 3 ⊢ (𝑁 = +∞ → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
23 | 2, 22 | jaoi 855 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∨ 𝑁 = +∞) → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
24 | 1, 23 | sylbi 216 | 1 ⊢ (𝑁 ∈ ℕ0* → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 class class class wbr 5153 0cc0 11158 1c1 11159 +∞cpnf 11295 ℝ*cxr 11297 ≤ cle 11299 2c2 12319 ℕ0cn0 12524 ℕ0*cxnn0 12596 |
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 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12597 |
This theorem is referenced by: vdgfrgrgt2 30231 xnn01gt 32674 lfuhgr2 34946 |
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