Theorem xnn0nn0d 33410

Description: Conditions for an extended nonnegative integer to be a nonnegative integer. (Contributed by Thierry Arnoux, 26-Oct-2025.)

Hypotheses

Ref	Expression
xnn0nnd.1	⊢ (𝜑 → 𝑁 ∈ ℕ0*)
xnn0nnd.2	⊢ (𝜑 → 𝑁 ∈ ℝ)

Assertion

Ref	Expression
xnn0nn0d	⊢ (𝜑 → 𝑁 ∈ ℕ0)

Proof of Theorem xnn0nn0d

Step	Hyp	Ref	Expression
1		xnn0nnd.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0*)
2		elxnn0 12684	. . 3 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞))
3	1, 2	sylib 221	. 2 ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞))
4		xnn0nnd.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
5	4	renepnfd 11363	. . 3 ⊢ (𝜑 → 𝑁 ≠ +∞)
6	5	neneqd 2960	. 2 ⊢ (𝜑 → ¬ 𝑁 = +∞)
7	3, 6	olcnd 891	1 ⊢ (𝜑 → 𝑁 ∈ ℕ0)

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ℝcr 11202  +∞cpnf 11343  ℕ₀cn0 12609  ℕ₀^*cxnn0 12682

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-un 7751  ax-cnex 11259  ax-resscn 11260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-nel 3062  df-rab 3413  df-v 3452  df-ss 3896  df-in 4091  df-un 4151  df-pw 4558  df-sn 4585  df-uni 4867  df-pnf 11348  df-xnn0 12683

This theorem is used by:  xnn0nnd  33411  constrext2chnlem  34428