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Theorem xnn0nnn0pnf 12561
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 12550 . . 3 (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0𝑁 = +∞))
2 pm2.53 847 . . 3 ((𝑁 ∈ ℕ0𝑁 = +∞) → (¬ 𝑁 ∈ ℕ0𝑁 = +∞))
31, 2sylbi 216 . 2 (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ ℕ0𝑁 = +∞))
43imp 405 1 ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 843   = wceq 1539  wcel 2104  +∞cpnf 11249  0cn0 12476  0*cxnn0 12548
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-pow 5362  ax-un 7727  ax-cnex 11168
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-un 3952  df-in 3954  df-ss 3964  df-pw 4603  df-sn 4628  df-uni 4908  df-pnf 11254  df-xnn0 12549
This theorem is referenced by:  xnn0xaddcl  13218  xnn0lem1lt  13227  nn0xmulclb  32251
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