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

Step	Hyp	Ref	Expression
1		elxnn0 12601	. . 3 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞))
2		pm2.53 852	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∨ 𝑁 = +∞) → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞))
3	1, 2	sylbi 217	. 2 ⊢ (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞))
4	3	imp 406	1 ⊢ ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  +∞cpnf 11292  ℕ₀cn0 12526  ℕ₀^*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