ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xnn0nnn0pnf GIF version

Theorem xnn0nnn0pnf 8501
Description: An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.)
Assertion
Ref Expression
xnn0nnn0pnf ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞)

Proof of Theorem xnn0nnn0pnf
StepHypRef Expression
1 elxnn0 8490 . . 3 (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0𝑁 = +∞))
2 pm2.53 674 . . 3 ((𝑁 ∈ ℕ0𝑁 = +∞) → (¬ 𝑁 ∈ ℕ0𝑁 = +∞))
31, 2sylbi 119 . 2 (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ ℕ0𝑁 = +∞))
43imp 122 1 ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662   = wceq 1285  wcel 1434  +∞cpnf 7282  0cn0 8425  0*cxnn0 8488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-un 4216  ax-cnex 7199
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-uni 3622  df-pnf 7287  df-xr 7289  df-xnn0 8489
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator