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Mirrors > Home > MPE Home > Th. List > xnn0nemnf | Structured version Visualization version GIF version |
Description: No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
xnn0nemnf | ⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxnn0 11779 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | |
2 | nn0re 11715 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
3 | 2 | renemnfd 10490 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ -∞) |
4 | pnfnemnf 10494 | . . . 4 ⊢ +∞ ≠ -∞ | |
5 | neeq1 3022 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 ≠ -∞ ↔ +∞ ≠ -∞)) | |
6 | 4, 5 | mpbiri 250 | . . 3 ⊢ (𝐴 = +∞ → 𝐴 ≠ -∞) |
7 | 3, 6 | jaoi 844 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 = +∞) → 𝐴 ≠ -∞) |
8 | 1, 7 | sylbi 209 | 1 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 834 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 +∞cpnf 10469 -∞cmnf 10470 ℕ0cn0 11705 ℕ0*cxnn0 11777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-i2m1 10401 ax-1ne0 10402 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-nn 11438 df-n0 11706 df-xnn0 11778 |
This theorem is referenced by: xnn0xrnemnf 11789 |
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