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Theorem xnn0nnn0pnf 9160
 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 9149 . . 3 (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0𝑁 = +∞))
2 pm2.53 712 . . 3 ((𝑁 ∈ ℕ0𝑁 = +∞) → (¬ 𝑁 ∈ ℕ0𝑁 = +∞))
31, 2sylbi 120 . 2 (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ ℕ0𝑁 = +∞))
43imp 123 1 ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698   = wceq 1335   ∈ wcel 2128  +∞cpnf 7903  ℕ0cn0 9084  ℕ0*cxnn0 9147 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-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-un 4393  ax-cnex 7817 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-pnf 7908  df-xr 7910  df-xnn0 9148 This theorem is referenced by: (None)
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