Theorem nnuni 36037

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 7869	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2		unieq 4873	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
3		uni0 4891	. . . . 5 ⊢ ∪ ∅ = ∅
4	2, 3	eqtrdi 2812	. . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
5		peano1 7863	. . . 4 ⊢ ∅ ∈ ω
6	4, 5	eqeltrdi 2869	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ∈ ω)
7		nnord 7848	. . . . . . 7 ⊢ (𝑥 ∈ ω → Ord 𝑥)
8		ordunisuc 7806	. . . . . . 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 2861	. . . . 5 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 ∈ ω)
12		unieq 4873	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = ∪ suc 𝑥)
13	12	eleq1d 2846	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (∪ 𝐴 ∈ ω ↔ ∪ suc 𝑥 ∈ ω))
14	11, 13	syl5ibrcom 249	. . . 4 ⊢ (𝑥 ∈ ω → (𝐴 = suc 𝑥 → ∪ 𝐴 ∈ ω))
15	14	rexlimiv 3155	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → ∪ 𝐴 ∈ ω)
16	6, 15	jaoi 868	. 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → ∪ 𝐴 ∈ ω)
17	1, 16	syl 17	1 ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)

Syntax hints:   → wi 4   ∨ wo 858   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  ∅c0 4283  ∪ cuni 4862  Ord word 6339  suc csuc 6342  ωcom 7840

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 5243  ax-nul 5253  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-tr 5205  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-om 7841

This theorem is referenced by: (None)