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Theorem nnuni 35727
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.
StepHypRef Expression
1 nn0suc 7916 . 2 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2 unieq 4918 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
3 uni0 4935 . . . . 5 ∅ = ∅
42, 3eqtrdi 2793 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
5 peano1 7910 . . . 4 ∅ ∈ ω
64, 5eqeltrdi 2849 . . 3 (𝐴 = ∅ → 𝐴 ∈ ω)
7 nnord 7895 . . . . . . 7 (𝑥 ∈ ω → Ord 𝑥)
8 ordunisuc 7852 . . . . . . 7 (Ord 𝑥 suc 𝑥 = 𝑥)
97, 8syl 17 . . . . . 6 (𝑥 ∈ ω → suc 𝑥 = 𝑥)
10 id 22 . . . . . 6 (𝑥 ∈ ω → 𝑥 ∈ ω)
119, 10eqeltrd 2841 . . . . 5 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
12 unieq 4918 . . . . . 6 (𝐴 = suc 𝑥 𝐴 = suc 𝑥)
1312eleq1d 2826 . . . . 5 (𝐴 = suc 𝑥 → ( 𝐴 ∈ ω ↔ suc 𝑥 ∈ ω))
1411, 13syl5ibrcom 247 . . . 4 (𝑥 ∈ ω → (𝐴 = suc 𝑥 𝐴 ∈ ω))
1514rexlimiv 3148 . . 3 (∃𝑥 ∈ ω 𝐴 = suc 𝑥 𝐴 ∈ ω)
166, 15jaoi 858 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
171, 16syl 17 1 (𝐴 ∈ ω → 𝐴 ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 848   = wceq 1540  wcel 2108  wrex 3070  c0 4333   cuni 4907  Ord word 6383  suc csuc 6386  ωcom 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-om 7888
This theorem is referenced by: (None)
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