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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 7916 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) | |
| 2 | unieq 4918 | . . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅) | |
| 3 | uni0 4935 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
| 4 | 2, 3 | eqtrdi 2793 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅) |
| 5 | peano1 7910 | . . . 4 ⊢ ∅ ∈ ω | |
| 6 | 4, 5 | eqeltrdi 2849 | . . 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 2841 | . . . . 5 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 ∈ ω) |
| 12 | unieq 4918 | . . . . . 6 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = ∪ suc 𝑥) | |
| 13 | 12 | eleq1d 2826 | . . . . 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 3148 | . . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → ∪ 𝐴 ∈ ω) |
| 16 | 6, 15 | jaoi 858 | . 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 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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