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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 12589 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | |
2 | nn0re 12524 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
3 | 2 | renemnfd 11304 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ -∞) |
4 | pnfnemnf 11307 | . . . 4 ⊢ +∞ ≠ -∞ | |
5 | neeq1 2993 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 ≠ -∞ ↔ +∞ ≠ -∞)) | |
6 | 4, 5 | mpbiri 257 | . . 3 ⊢ (𝐴 = +∞ → 𝐴 ≠ -∞) |
7 | 3, 6 | jaoi 855 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 = +∞) → 𝐴 ≠ -∞) |
8 | 1, 7 | sylbi 216 | 1 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 +∞cpnf 11283 -∞cmnf 11284 ℕ0cn0 12515 ℕ0*cxnn0 12587 |
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 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-i2m1 11214 ax-1ne0 11215 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7416 df-om 7866 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-nn 12256 df-n0 12516 df-xnn0 12588 |
This theorem is referenced by: xnn0xrnemnf 12599 |
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