Theorem nnuni 35769

Description: The union of a finite ordinal is a finite ordinal. (Contributed by Scott Fenton, 17-Oct-2024.)

Assertion

Ref	Expression
nnuni	⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)

Proof of Theorem nnuni

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0suc 7824	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2		unieq 4870	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
3		uni0 4887	. . . . 5 ⊢ ∪ ∅ = ∅
4	2, 3	eqtrdi 2782	. . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
5		peano1 7819	. . . 4 ⊢ ∅ ∈ ω
6	4, 5	eqeltrdi 2839	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ∈ ω)
7		nnord 7804	. . . . . . 7 ⊢ (𝑥 ∈ ω → Ord 𝑥)
8		ordunisuc 7762	. . . . . . 7 ⊢ (Ord 𝑥 → ∪ suc 𝑥 = 𝑥)
9	7, 8	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 = 𝑥)
10		id 22	. . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ∈ ω)
11	9, 10	eqeltrd 2831	. . . . 5 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 ∈ ω)
12		unieq 4870	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = ∪ suc 𝑥)
13	12	eleq1d 2816	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (∪ 𝐴 ∈ ω ↔ ∪ suc 𝑥 ∈ ω))
14	11, 13	syl5ibrcom 247	. . . 4 ⊢ (𝑥 ∈ ω → (𝐴 = suc 𝑥 → ∪ 𝐴 ∈ ω))
15	14	rexlimiv 3126	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → ∪ 𝐴 ∈ ω)
16	6, 15	jaoi 857	. 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∪ 𝐴 ∈ ω)
17	1, 16	syl 17	1 ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  ∅c0 4283  ∪ cuni 4859  Ord word 6305  suc csuc 6308  ωcom 7796

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-11 2160  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-om 7797

This theorem is referenced by: (None)