Theorem nnuni 36471

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 7904	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2		unieq 4878	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
3		uni0 4896	. . . . 5 ⊢ ∪ ∅ = ∅
4	2, 3	eqtrdi 2812	. . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
5		peano1 7898	. . . 4 ⊢ ∅ ∈ ω
6	4, 5	eqeltrdi 2869	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ∈ ω)
7		nnord 7883	. . . . . . 7 ⊢ (𝑥 ∈ ω → Ord 𝑥)
8		ordunisuc 7841	. . . . . . 7 ⊢ (Ord 𝑥 → ∪ suc 𝑥 = 𝑥)
9	7, 8	syl 18	. . . . . 6 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 = 𝑥)
10		id 23	. . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ∈ ω)
11	9, 10	eqeltrd 2861	. . . . 5 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 ∈ ω)
12		unieq 4878	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = ∪ suc 𝑥)
13	12	eleq1d 2846	. . . . 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 3157	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → ∪ 𝐴 ∈ ω)
16	6, 15	jaoi 871	. 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∪ 𝐴 ∈ ω)
17	1, 16	syl 18	1 ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  ∅c0 4279  ∪ cuni 4867  Ord word 6360  suc csuc 6363  ωcom 7875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-om 7876

This theorem is used by: (None)