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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 7831 | . . . . 5 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 2406 | . . . 4 ⊢ ¬ +∞ ∈ ℝ |
3 | nn0re 9010 | . . . 4 ⊢ (+∞ ∈ ℕ0 → +∞ ∈ ℝ) | |
4 | 2, 3 | mto 652 | . . 3 ⊢ ¬ +∞ ∈ ℕ0 |
5 | eleq1 2203 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℕ0 ↔ +∞ ∈ ℕ0)) | |
6 | 4, 5 | mtbiri 665 | . 2 ⊢ (𝐴 = +∞ → ¬ 𝐴 ∈ ℕ0) |
7 | 6 | necon2ai 2363 | 1 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 ≠ wne 2309 ℝcr 7643 +∞cpnf 7821 ℕ0cn0 9001 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-un 4363 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 ax-rnegex 7753 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-uni 3745 df-int 3780 df-pnf 7826 df-inn 8745 df-n0 9002 |
This theorem is referenced by: nn0nepnfd 9074 fxnn0nninf 10242 0tonninf 10243 1tonninf 10244 |
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