![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12562 | . 2 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) | |
2 | nn0n0n1ge2b 12556 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) | |
3 | 0nn0 12503 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
4 | nn0nepnf 12568 | . . . . . . . 8 ⊢ (0 ∈ ℕ0 → 0 ≠ +∞) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 0 ≠ +∞ |
6 | 5 | necomi 2990 | . . . . . 6 ⊢ +∞ ≠ 0 |
7 | neeq1 2998 | . . . . . 6 ⊢ (𝑁 = +∞ → (𝑁 ≠ 0 ↔ +∞ ≠ 0)) | |
8 | 6, 7 | mpbiri 258 | . . . . 5 ⊢ (𝑁 = +∞ → 𝑁 ≠ 0) |
9 | 1nn0 12504 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
10 | nn0nepnf 12568 | . . . . . . . 8 ⊢ (1 ∈ ℕ0 → 1 ≠ +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ 1 ≠ +∞ |
12 | 11 | necomi 2990 | . . . . . 6 ⊢ +∞ ≠ 1 |
13 | neeq1 2998 | . . . . . 6 ⊢ (𝑁 = +∞ → (𝑁 ≠ 1 ↔ +∞ ≠ 1)) | |
14 | 12, 13 | mpbiri 258 | . . . . 5 ⊢ (𝑁 = +∞ → 𝑁 ≠ 1) |
15 | 8, 14 | jca 511 | . . . 4 ⊢ (𝑁 = +∞ → (𝑁 ≠ 0 ∧ 𝑁 ≠ 1)) |
16 | 2re 12302 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
17 | 16 | rexri 11288 | . . . . . 6 ⊢ 2 ∈ ℝ* |
18 | pnfge 13128 | . . . . . 6 ⊢ (2 ∈ ℝ* → 2 ≤ +∞) | |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ 2 ≤ +∞ |
20 | breq2 5146 | . . . . 5 ⊢ (𝑁 = +∞ → (2 ≤ 𝑁 ↔ 2 ≤ +∞)) | |
21 | 19, 20 | mpbiri 258 | . . . 4 ⊢ (𝑁 = +∞ → 2 ≤ 𝑁) |
22 | 15, 21 | 2thd 265 | . . 3 ⊢ (𝑁 = +∞ → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) |
23 | 2, 22 | jaoi 856 | . 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 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 0cc0 11124 1c1 11125 +∞cpnf 11261 ℝ*cxr 11263 ≤ cle 11265 2c2 12283 ℕ0cn0 12488 ℕ0*cxnn0 12560 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-n0 12489 df-xnn0 12561 |
This theorem is referenced by: vdgfrgrgt2 30082 xnn01gt 32511 lfuhgr2 34651 |
Copyright terms: Public domain | W3C validator |