Theorem nnuni 33405

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 7670	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2		unieq 4827	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
3		uni0 4846	. . . . 5 ⊢ ∪ ∅ = ∅
4	2, 3	eqtrdi 2794	. . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
5		peano1 7664	. . . 4 ⊢ ∅ ∈ ω
6	4, 5	eqeltrdi 2846	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ∈ ω)
7		nnord 7649	. . . . . . 7 ⊢ (𝑥 ∈ ω → Ord 𝑥)
8		ordunisuc 7608	. . . . . . 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 2838	. . . . 5 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 ∈ ω)
12		unieq 4827	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = ∪ suc 𝑥)
13	12	eleq1d 2822	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (∪ 𝐴 ∈ ω ↔ ∪ suc 𝑥 ∈ ω))
14	11, 13	syl5ibrcom 250	. . . 4 ⊢ (𝑥 ∈ ω → (𝐴 = suc 𝑥 → ∪ 𝐴 ∈ ω))
15	14	rexlimiv 3196	. . 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 1543   ∈ wcel 2110  ∃wrex 3059  ∅c0 4234  ∪ cuni 4816  Ord word 6209  suc csuc 6212  ωcom 7641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2708  ax-sep 5189  ax-nul 5196  ax-pr 5319  ax-un 7520

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2940  df-ral 3063  df-rex 3064  df-rab 3067  df-v 3407  df-dif 3866  df-un 3868  df-in 3870  df-ss 3880  df-pss 3882  df-nul 4235  df-if 4437  df-pw 4512  df-sn 4539  df-pr 4541  df-tp 4543  df-op 4545  df-uni 4817  df-br 5051  df-opab 5113  df-tr 5159  df-eprel 5457  df-po 5465  df-so 5466  df-fr 5506  df-we 5508  df-ord 6213  df-on 6214  df-lim 6215  df-suc 6216  df-om 7642

This theorem is referenced by: (None)