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Mirrors > Home > ILE Home > Th. List > nn0nepnf | 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 7775 | . . . . 5 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 2382 | . . . 4 ⊢ ¬ +∞ ∈ ℝ |
3 | nn0re 8954 | . . . 4 ⊢ (+∞ ∈ ℕ0 → +∞ ∈ ℝ) | |
4 | 2, 3 | mto 636 | . . 3 ⊢ ¬ +∞ ∈ ℕ0 |
5 | eleq1 2180 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℕ0 ↔ +∞ ∈ ℕ0)) | |
6 | 4, 5 | mtbiri 649 | . 2 ⊢ (𝐴 = +∞ → ¬ 𝐴 ∈ ℕ0) |
7 | 6 | necon2ai 2339 | 1 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 ≠ wne 2285 ℝcr 7587 +∞cpnf 7765 ℕ0cn0 8945 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-un 4325 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 ax-rnegex 7697 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-uni 3707 df-int 3742 df-pnf 7770 df-inn 8689 df-n0 8946 |
This theorem is referenced by: nn0nepnfd 9018 fxnn0nninf 10179 0tonninf 10180 1tonninf 10181 |
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