Theorem xnn0nnn0pnf 11983

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

Step	Hyp	Ref	Expression
1		elxnn0 11972	. . 3 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞))
2		pm2.53 847	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∨ 𝑁 = +∞) → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞))
3	1, 2	sylbi 219	. 2 ⊢ (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞))
4	3	imp 409	1 ⊢ ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1536   ∈ wcel 2113  +∞cpnf 10675  ℕ₀cn0 11900  ℕ₀^*cxnn0 11970

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-pow 5269  ax-un 7464  ax-cnex 10596

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-un 3944  df-in 3946  df-ss 3955  df-pw 4544  df-sn 4571  df-uni 4842  df-pnf 10680  df-xnn0 11971

This theorem is referenced by:  xnn0xaddcl  12631  xnn0lem1lt  12640  nn0xmulclb  30499