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Mirrors > Home > MPE Home > Th. List > ssorduni | Structured version Visualization version GIF version |
Description: The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
ssorduni | ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni2 4795 | . . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
2 | ssel 3881 | . . . . . . . . 9 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → 𝑦 ∈ On)) | |
3 | onelss 6204 | . . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦)) | |
4 | 2, 3 | syl6 35 | . . . . . . . 8 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))) |
5 | anc2r 559 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦)) → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → (𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴)))) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → (𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴)))) |
7 | ssuni 4818 | . . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ⊆ ∪ 𝐴) | |
8 | 6, 7 | syl8 76 | . . . . . 6 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴))) |
9 | 8 | rexlimdv 3205 | . . . . 5 ⊢ (𝐴 ⊆ On → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴)) |
10 | 1, 9 | syl5bi 245 | . . . 4 ⊢ (𝐴 ⊆ On → (𝑥 ∈ ∪ 𝐴 → 𝑥 ⊆ ∪ 𝐴)) |
11 | 10 | ralrimiv 3110 | . . 3 ⊢ (𝐴 ⊆ On → ∀𝑥 ∈ ∪ 𝐴𝑥 ⊆ ∪ 𝐴) |
12 | dftr3 5135 | . . 3 ⊢ (Tr ∪ 𝐴 ↔ ∀𝑥 ∈ ∪ 𝐴𝑥 ⊆ ∪ 𝐴) | |
13 | 11, 12 | sylibr 237 | . 2 ⊢ (𝐴 ⊆ On → Tr ∪ 𝐴) |
14 | onelon 6187 | . . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On) | |
15 | 14 | ex 417 | . . . . . 6 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ∈ On)) |
16 | 2, 15 | syl6 35 | . . . . 5 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ∈ On))) |
17 | 16 | rexlimdv 3205 | . . . 4 ⊢ (𝐴 ⊆ On → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ∈ On)) |
18 | 1, 17 | syl5bi 245 | . . 3 ⊢ (𝐴 ⊆ On → (𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ On)) |
19 | 18 | ssrdv 3894 | . 2 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ⊆ On) |
20 | ordon 7490 | . . 3 ⊢ Ord On | |
21 | trssord 6179 | . . . 4 ⊢ ((Tr ∪ 𝐴 ∧ ∪ 𝐴 ⊆ On ∧ Ord On) → Ord ∪ 𝐴) | |
22 | 21 | 3exp 1117 | . . 3 ⊢ (Tr ∪ 𝐴 → (∪ 𝐴 ⊆ On → (Ord On → Ord ∪ 𝐴))) |
23 | 20, 22 | mpii 46 | . 2 ⊢ (Tr ∪ 𝐴 → (∪ 𝐴 ⊆ On → Ord ∪ 𝐴)) |
24 | 13, 19, 23 | sylc 65 | 1 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 ∈ wcel 2112 ∀wral 3068 ∃wrex 3069 ⊆ wss 3854 ∪ cuni 4791 Tr wtr 5131 Ord word 6161 Oncon0 6162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-tr 5132 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-ord 6165 df-on 6166 |
This theorem is referenced by: ssonuni 7493 ssonprc 7499 orduni 7501 onsucuni 7535 limuni3 7559 onfununi 7981 tfrlem8 8023 onssnum 9485 unialeph 9546 cfslbn 9712 hsmexlem1 9871 inaprc 10281 bdayimaon 33446 noetasuplem4 33489 noetainflem4 33493 noeta2 33529 etasslt2 33554 scutbdaybnd2lim 33557 |
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