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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 9194 | . . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴) | |
| 2 | suceq 6382 | . . . 4 ⊢ (𝑥 = ∪ 𝐴 → suc 𝑥 = suc ∪ 𝐴) | |
| 3 | 2 | ssiun2s 4981 | . . 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 7726 | . . . . . 6 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
| 6 | ordsuci 7755 | . . . . . 6 ⊢ (Ord ∪ 𝐴 → Ord suc ∪ 𝐴) | |
| 7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝐴 ⊆ On → Ord suc ∪ 𝐴) |
| 8 | onsucuni 7772 | . . . . . 6 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) | |
| 9 | 8 | sselda 3917 | . . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ suc ∪ 𝐴) |
| 10 | ordsucss 7762 | . . . . . 6 ⊢ (Ord suc ∪ 𝐴 → (𝑥 ∈ suc ∪ 𝐴 → suc 𝑥 ⊆ suc ∪ 𝐴)) | |
| 11 | 10 | imp 408 | . . . . 5 ⊢ ((Ord suc ∪ 𝐴 ∧ 𝑥 ∈ suc ∪ 𝐴) → suc 𝑥 ⊆ suc ∪ 𝐴) |
| 12 | 7, 9, 11 | syl2an2r 692 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ⊆ suc ∪ 𝐴) |
| 13 | 12 | iunssd 4983 | . . 3 ⊢ (𝐴 ⊆ On → ∪ 𝑥 ∈ 𝐴 suc 𝑥 ⊆ suc ∪ 𝐴) |
| 14 | 13 | 3ad2ant1 1140 | . 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 suc 𝑥 ⊆ suc ∪ 𝐴) |
| 15 | 4, 14 | eqssd 3934 | 1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → suc ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 suc 𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ⊆ wss 3885 ∅c0 4264 ∪ cuni 4841 ∪ ciun 4924 Ord word 6313 Oncon0 6314 suc csuc 6316 Fincfn 8887 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pr 5365 ax-un 7682 |
| This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-om 7811 df-en 8888 df-fin 8891 |
| This theorem is referenced by: (None) |
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