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Theorem xnn0nnn0pnf 12612
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 12601 . . 3 (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0𝑁 = +∞))
2 pm2.53 852 . . 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 848   = wceq 1540  wcel 2108  +∞cpnf 11292  0cn0 12526  0*cxnn0 12599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-pow 5365  ax-un 7755  ax-cnex 11211
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-pw 4602  df-sn 4627  df-uni 4908  df-pnf 11297  df-xnn0 12600
This theorem is referenced by:  xnn0xaddcl  13277  xnn0lem1lt  13286  nn0xmulclb  32775
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