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| Mirrors > Home > MPE Home > Th. List > Mathboxes > onsucunifi | Structured version Visualization version GIF version | ||
| Description: The successor to the union of any non-empty, finite subset of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| onsucunifi | ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → suc ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 suc 𝑥) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ordunifi 9326 | . . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴) | |
| 2 | suceq 6450 | . . . 4 ⊢ (𝑥 = ∪ 𝐴 → suc 𝑥 = suc ∪ 𝐴) | |
| 3 | 2 | ssiun2s 5048 | . . 3 ⊢ (∪ 𝐴 ∈ 𝐴 → suc ∪ 𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 suc 𝑥) | 
| 4 | 1, 3 | syl 17 | . 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → suc ∪ 𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 suc 𝑥) | 
| 5 | ssorduni 7799 | . . . . . 6 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
| 6 | ordsuci 7828 | . . . . . 6 ⊢ (Ord ∪ 𝐴 → Ord suc ∪ 𝐴) | |
| 7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝐴 ⊆ On → Ord suc ∪ 𝐴) | 
| 8 | onsucuni 7848 | . . . . . 6 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) | |
| 9 | 8 | sselda 3983 | . . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ suc ∪ 𝐴) | 
| 10 | ordsucss 7838 | . . . . . 6 ⊢ (Ord suc ∪ 𝐴 → (𝑥 ∈ suc ∪ 𝐴 → suc 𝑥 ⊆ suc ∪ 𝐴)) | |
| 11 | 10 | imp 406 | . . . . 5 ⊢ ((Ord suc ∪ 𝐴 ∧ 𝑥 ∈ suc ∪ 𝐴) → suc 𝑥 ⊆ suc ∪ 𝐴) | 
| 12 | 7, 9, 11 | syl2an2r 685 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ⊆ suc ∪ 𝐴) | 
| 13 | 12 | iunssd 5050 | . . 3 ⊢ (𝐴 ⊆ On → ∪ 𝑥 ∈ 𝐴 suc 𝑥 ⊆ suc ∪ 𝐴) | 
| 14 | 13 | 3ad2ant1 1134 | . 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 suc 𝑥 ⊆ suc ∪ 𝐴) | 
| 15 | 4, 14 | eqssd 4001 | 1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → suc ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 suc 𝑥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 ∪ ciun 4991 Ord word 6383 Oncon0 6384 suc csuc 6386 Fincfn 8985 | 
| 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-10 2141 ax-11 2157 ax-12 2177 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 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-iun 4993 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-en 8986 df-fin 8989 | 
| This theorem is referenced by: (None) | 
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