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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 7917 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) | |
2 | unieq 4923 | . . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅) | |
3 | uni0 4940 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
4 | 2, 3 | eqtrdi 2791 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅) |
5 | peano1 7911 | . . . 4 ⊢ ∅ ∈ ω | |
6 | 4, 5 | eqeltrdi 2847 | . . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ∈ ω) |
7 | nnord 7895 | . . . . . . 7 ⊢ (𝑥 ∈ ω → Ord 𝑥) | |
8 | ordunisuc 7852 | . . . . . . 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 2839 | . . . . 5 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 ∈ ω) |
12 | unieq 4923 | . . . . . 6 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = ∪ suc 𝑥) | |
13 | 12 | eleq1d 2824 | . . . . 5 ⊢ (𝐴 = suc 𝑥 → (∪ 𝐴 ∈ ω ↔ ∪ suc 𝑥 ∈ ω)) |
14 | 11, 13 | syl5ibrcom 247 | . . . 4 ⊢ (𝑥 ∈ ω → (𝐴 = suc 𝑥 → ∪ 𝐴 ∈ ω)) |
15 | 14 | rexlimiv 3146 | . . 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 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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