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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnuni | Structured version Visualization version GIF version |
Description: The union of a finite ordinal is a finite ordinal. (Contributed by Scott Fenton, 17-Oct-2024.) |
Ref | Expression |
---|---|
nnuni | ⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω) |
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) |
Colors of variables: wff setvar class |
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) |
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