![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
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 9295 | . . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴) | |
2 | suceq 6424 | . . . 4 ⊢ (𝑥 = ∪ 𝐴 → suc 𝑥 = suc ∪ 𝐴) | |
3 | 2 | ssiun2s 5044 | . . 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 7763 | . . . . . 6 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
6 | ordsuci 7793 | . . . . . 6 ⊢ (Ord ∪ 𝐴 → Ord suc ∪ 𝐴) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝐴 ⊆ On → Ord suc ∪ 𝐴) |
8 | onsucuni 7813 | . . . . . 6 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) | |
9 | 8 | sselda 3977 | . . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ suc ∪ 𝐴) |
10 | ordsucss 7803 | . . . . . 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 682 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ⊆ suc ∪ 𝐴) |
13 | 12 | iunssd 5046 | . . 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 3994 | 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 2934 ⊆ wss 3943 ∅c0 4317 ∪ cuni 4902 ∪ ciun 4990 Ord word 6357 Oncon0 6358 suc csuc 6360 Fincfn 8941 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-om 7853 df-en 8942 df-fin 8945 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |