MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xnn0nnn0pnf Structured version   Visualization version   GIF version

Theorem xnn0nnn0pnf 12638
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 12627 . . 3 (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0𝑁 = +∞))
2 pm2.53 850 . . 3 ((𝑁 ∈ ℕ0𝑁 = +∞) → (¬ 𝑁 ∈ ℕ0𝑁 = +∞))
31, 2sylbi 217 . 2 (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ ℕ0𝑁 = +∞))
43imp 406 1 ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 846   = wceq 1537  wcel 2108  +∞cpnf 11321  0cn0 12553  0*cxnn0 12625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-pow 5383  ax-un 7770  ax-cnex 11240
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-ss 3993  df-pw 4624  df-sn 4649  df-uni 4932  df-pnf 11326  df-xnn0 12626
This theorem is referenced by:  xnn0xaddcl  13297  xnn0lem1lt  13306  nn0xmulclb  32778
  Copyright terms: Public domain W3C validator