Theorem xnn0nn0d 32845

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 12512	. . 3 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞))
3	1, 2	sylib 218	. 2 ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞))
4		xnn0nnd.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
5	4	renepnfd 11196	. . 3 ⊢ (𝜑 → 𝑁 ≠ +∞)
6	5	neneqd 2937	. 2 ⊢ (𝜑 → ¬ 𝑁 = +∞)
7	3, 6	olcnd 878	1 ⊢ (𝜑 → 𝑁 ∈ ℕ0)

Syntax hints:   → wi 4   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ℝcr 11037  +∞cpnf 11176  ℕ₀cn0 12437  ℕ₀^*cxnn0 12510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-nel 3037  df-rab 3390  df-v 3431  df-un 3894  df-in 3896  df-ss 3906  df-pw 4543  df-sn 4568  df-pr 4570  df-uni 4851  df-pnf 11181  df-xnn0 12511

This theorem is referenced by:  xnn0nnd  32846  constrext2chnlem  33894