![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nn0nepnf | Structured version Visualization version GIF version |
Description: No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
nn0nepnf | ⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfnre 10418 | . . . . 5 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 3076 | . . . 4 ⊢ ¬ +∞ ∈ ℝ |
3 | nn0re 11652 | . . . 4 ⊢ (+∞ ∈ ℕ0 → +∞ ∈ ℝ) | |
4 | 2, 3 | mto 189 | . . 3 ⊢ ¬ +∞ ∈ ℕ0 |
5 | eleq1 2846 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℕ0 ↔ +∞ ∈ ℕ0)) | |
6 | 4, 5 | mtbiri 319 | . 2 ⊢ (𝐴 = +∞ → ¬ 𝐴 ∈ ℕ0) |
7 | 6 | necon2ai 2997 | 1 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ℝcr 10271 +∞cpnf 10408 ℕ0cn0 11642 |
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-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-i2m1 10340 ax-1ne0 10341 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 |
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-ov 6925 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-pnf 10413 df-nn 11375 df-n0 11643 |
This theorem is referenced by: nn0nepnfd 11724 xnn0n0n1ge2b 12276 |
Copyright terms: Public domain | W3C validator |