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Theorem nnuni 35707
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 7917 . 2 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2 unieq 4923 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
3 uni0 4940 . . . . 5 ∅ = ∅
42, 3eqtrdi 2791 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
5 peano1 7911 . . . 4 ∅ ∈ ω
64, 5eqeltrdi 2847 . . 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 2839 . . . . 5 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
12 unieq 4923 . . . . . 6 (𝐴 = suc 𝑥 𝐴 = suc 𝑥)
1312eleq1d 2824 . . . . 5 (𝐴 = suc 𝑥 → ( 𝐴 ∈ ω ↔ suc 𝑥 ∈ ω))
1411, 13syl5ibrcom 247 . . . 4 (𝑥 ∈ ω → (𝐴 = suc 𝑥 𝐴 ∈ ω))
1514rexlimiv 3146 . . 3 (∃𝑥 ∈ ω 𝐴 = suc 𝑥 𝐴 ∈ ω)
166, 15jaoi 857 . 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 847   = wceq 1537  wcel 2106  wrex 3068  c0 4339   cuni 4912  Ord word 6385  suc csuc 6388  ωcom 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-om 7888
This theorem is referenced by: (None)
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