Theorem xnn0nn0d 32924

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 12553	. . 3 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞))
3	1, 2	sylib 220	. 2 ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞))
4		xnn0nnd.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
5	4	renepnfd 11230	. . 3 ⊢ (𝜑 → 𝑁 ≠ +∞)
6	5	neneqd 2961	. 2 ⊢ (𝜑 → ¬ 𝑁 = +∞)
7	3, 6	olcnd 888	1 ⊢ (𝜑 → 𝑁 ∈ ℕ0)

Syntax hints:   → wi 4   ∨ wo 858   = wceq 1559   ∈ wcel 2141  ℝcr 11069  +∞cpnf 11210  ℕ₀cn0 12478  ℕ₀^*cxnn0 12551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pow 5321  ax-un 7714  ax-cnex 11126  ax-resscn 11127

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-nel 3061  df-rab 3414  df-v 3455  df-un 3909  df-in 3911  df-ss 3921  df-pw 4556  df-sn 4582  df-uni 4865  df-pnf 11215  df-xnn0 12552

This theorem is referenced by:  xnn0nnd  32925  constrext2chnlem  34008