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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 11963 | . 2 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) | |
2 | nn0n0n1ge2b 11957 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) | |
3 | 0nn0 11906 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
4 | nn0nepnf 11969 | . . . . . . . 8 ⊢ (0 ∈ ℕ0 → 0 ≠ +∞) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 0 ≠ +∞ |
6 | 5 | necomi 3070 | . . . . . 6 ⊢ +∞ ≠ 0 |
7 | neeq1 3078 | . . . . . 6 ⊢ (𝑁 = +∞ → (𝑁 ≠ 0 ↔ +∞ ≠ 0)) | |
8 | 6, 7 | mpbiri 260 | . . . . 5 ⊢ (𝑁 = +∞ → 𝑁 ≠ 0) |
9 | 1nn0 11907 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
10 | nn0nepnf 11969 | . . . . . . . 8 ⊢ (1 ∈ ℕ0 → 1 ≠ +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ 1 ≠ +∞ |
12 | 11 | necomi 3070 | . . . . . 6 ⊢ +∞ ≠ 1 |
13 | neeq1 3078 | . . . . . 6 ⊢ (𝑁 = +∞ → (𝑁 ≠ 1 ↔ +∞ ≠ 1)) | |
14 | 12, 13 | mpbiri 260 | . . . . 5 ⊢ (𝑁 = +∞ → 𝑁 ≠ 1) |
15 | 8, 14 | jca 514 | . . . 4 ⊢ (𝑁 = +∞ → (𝑁 ≠ 0 ∧ 𝑁 ≠ 1)) |
16 | 2re 11705 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
17 | 16 | rexri 10693 | . . . . . 6 ⊢ 2 ∈ ℝ* |
18 | pnfge 12519 | . . . . . 6 ⊢ (2 ∈ ℝ* → 2 ≤ +∞) | |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ 2 ≤ +∞ |
20 | breq2 5063 | . . . . 5 ⊢ (𝑁 = +∞ → (2 ≤ 𝑁 ↔ 2 ≤ +∞)) | |
21 | 19, 20 | mpbiri 260 | . . . 4 ⊢ (𝑁 = +∞ → 2 ≤ 𝑁) |
22 | 15, 21 | 2thd 267 | . . 3 ⊢ (𝑁 = +∞ → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
23 | 2, 22 | jaoi 853 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∨ 𝑁 = +∞) → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
24 | 1, 23 | sylbi 219 | 1 ⊢ (𝑁 ∈ ℕ0* → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5059 0cc0 10531 1c1 10532 +∞cpnf 10666 ℝ*cxr 10668 ≤ cle 10670 2c2 11686 ℕ0cn0 11891 ℕ0*cxnn0 11961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-xnn0 11962 |
This theorem is referenced by: vdgfrgrgt2 28071 xnn01gt 30489 lfuhgr2 32360 |
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