Theorem xnn0nn0d 33128

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 12585	. . 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 11266	. . 3 ⊢ (𝜑 → 𝑁 ≠ +∞)
6	5	neneqd 2962	. 2 ⊢ (𝜑 → ¬ 𝑁 = +∞)
7	3, 6	olcnd 890	1 ⊢ (𝜑 → 𝑁 ∈ ℕ0)

This proof depends on syntax axioms:   → wi 4   ∨ wo 860   = wceq 1569   ∈ wcel 2142  ℝcr 11105  +∞cpnf 11246  ℕ₀cn0 12510  ℕ₀^*cxnn0 12583

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pow 5335  ax-un 7734  ax-cnex 11162  ax-resscn 11163

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-nel 3064  df-rab 3416  df-v 3456  df-un 3909  df-in 3911  df-ss 3921  df-pw 4563  df-sn 4589  df-uni 4872  df-pnf 11251  df-xnn0 12584

This theorem is used by:  xnn0nnd  33129  constrext2chnlem  34149