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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 9314 | . . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴) | |
2 | suceq 6430 | . . . 4 ⊢ (𝑥 = ∪ 𝐴 → suc 𝑥 = suc ∪ 𝐴) | |
3 | 2 | ssiun2s 5046 | . . 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 7778 | . . . . . 6 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
6 | ordsuci 7808 | . . . . . 6 ⊢ (Ord ∪ 𝐴 → Ord suc ∪ 𝐴) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝐴 ⊆ On → Ord suc ∪ 𝐴) |
8 | onsucuni 7828 | . . . . . 6 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) | |
9 | 8 | sselda 3972 | . . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ suc ∪ 𝐴) |
10 | ordsucss 7818 | . . . . . 6 ⊢ (Ord suc ∪ 𝐴 → (𝑥 ∈ suc ∪ 𝐴 → suc 𝑥 ⊆ suc ∪ 𝐴)) | |
11 | 10 | imp 405 | . . . . 5 ⊢ ((Ord suc ∪ 𝐴 ∧ 𝑥 ∈ suc ∪ 𝐴) → suc 𝑥 ⊆ suc ∪ 𝐴) |
12 | 7, 9, 11 | syl2an2r 683 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ⊆ suc ∪ 𝐴) |
13 | 12 | iunssd 5048 | . . 3 ⊢ (𝐴 ⊆ On → ∪ 𝑥 ∈ 𝐴 suc 𝑥 ⊆ suc ∪ 𝐴) |
14 | 13 | 3ad2ant1 1130 | . 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 suc 𝑥 ⊆ suc ∪ 𝐴) |
15 | 4, 14 | eqssd 3990 | 1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → suc ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 suc 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ⊆ wss 3940 ∅c0 4318 ∪ cuni 4903 ∪ ciun 4991 Ord word 6363 Oncon0 6364 suc csuc 6366 Fincfn 8960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7868 df-en 8961 df-fin 8964 |
This theorem is referenced by: (None) |
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