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Mirrors > Home > ILE Home > Th. List > xnn0nemnf | 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 9035 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | |
2 | nn0re 8979 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
3 | 2 | renemnfd 7810 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ -∞) |
4 | pnfnemnf 7813 | . . . 4 ⊢ +∞ ≠ -∞ | |
5 | neeq1 2319 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 ≠ -∞ ↔ +∞ ≠ -∞)) | |
6 | 4, 5 | mpbiri 167 | . . 3 ⊢ (𝐴 = +∞ → 𝐴 ≠ -∞) |
7 | 3, 6 | jaoi 705 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 = +∞) → 𝐴 ≠ -∞) |
8 | 1, 7 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ≠ wne 2306 +∞cpnf 7790 -∞cmnf 7791 ℕ0cn0 8970 ℕ0*cxnn0 9033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 ax-rnegex 7722 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-pnf 7795 df-mnf 7796 df-xr 7797 df-inn 8714 df-n0 8971 df-xnn0 9034 |
This theorem is referenced by: xnn0xrnemnf 9045 |
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