Theorem xnn0nn0d 33098

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 12580	. . 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 11261	. . 3 ⊢ (𝜑 → 𝑁 ≠ +∞)
6	5	neneqd 2963	. 2 ⊢ (𝜑 → ¬ 𝑁 = +∞)
7	3, 6	olcnd 890	1 ⊢ (𝜑 → 𝑁 ∈ ℕ0)

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ℝcr 11100  +∞cpnf 11241  ℕ₀cn0 12505  ℕ₀^*cxnn0 12578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pow 5338  ax-un 7734  ax-cnex 11157  ax-resscn 11158

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-nel 3065  df-rab 3417  df-v 3457  df-un 3911  df-in 3913  df-ss 3923  df-pw 4565  df-sn 4591  df-uni 4874  df-pnf 11246  df-xnn0 12579

This theorem is referenced by:  xnn0nnd  33099  constrext2chnlem  34121