Theorem xnn0nn0d 32750

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 12453	. . 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 11160	. . 3 ⊢ (𝜑 → 𝑁 ≠ +∞)
6	5	neneqd 2933	. 2 ⊢ (𝜑 → ¬ 𝑁 = +∞)
7	3, 6	olcnd 877	1 ⊢ (𝜑 → 𝑁 ∈ ℕ0)

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ℝcr 11002  +∞cpnf 11140  ℕ₀cn0 12378  ℕ₀^*cxnn0 12451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-nel 3033  df-rab 3396  df-v 3438  df-un 3907  df-in 3909  df-ss 3919  df-pw 4552  df-sn 4577  df-pr 4579  df-uni 4860  df-pnf 11145  df-xnn0 12452

This theorem is referenced by:  xnn0nnd  32751  constrext2chnlem  33758