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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. Lemma 2.7 of [Schloeder] p. 4. (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 4913 | . . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
2 | ssel 3976 | . . . . . . . . 9 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → 𝑦 ∈ On)) | |
3 | onelss 6407 | . . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦)) | |
4 | 2, 3 | syl6 35 | . . . . . . . 8 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))) |
5 | anc2r 556 | . . . . . . . 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 4937 | . . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ⊆ ∪ 𝐴) | |
8 | 6, 7 | syl8 76 | . . . . . 6 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴))) |
9 | 8 | rexlimdv 3154 | . . . . 5 ⊢ (𝐴 ⊆ On → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴)) |
10 | 1, 9 | biimtrid 241 | . . . 4 ⊢ (𝐴 ⊆ On → (𝑥 ∈ ∪ 𝐴 → 𝑥 ⊆ ∪ 𝐴)) |
11 | 10 | ralrimiv 3146 | . . 3 ⊢ (𝐴 ⊆ On → ∀𝑥 ∈ ∪ 𝐴𝑥 ⊆ ∪ 𝐴) |
12 | dftr3 5272 | . . 3 ⊢ (Tr ∪ 𝐴 ↔ ∀𝑥 ∈ ∪ 𝐴𝑥 ⊆ ∪ 𝐴) | |
13 | 11, 12 | sylibr 233 | . 2 ⊢ (𝐴 ⊆ On → Tr ∪ 𝐴) |
14 | onelon 6390 | . . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On) | |
15 | 14 | ex 414 | . . . . . 6 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ∈ On)) |
16 | 2, 15 | syl6 35 | . . . . 5 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ∈ On))) |
17 | 16 | rexlimdv 3154 | . . . 4 ⊢ (𝐴 ⊆ On → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ∈ On)) |
18 | 1, 17 | biimtrid 241 | . . 3 ⊢ (𝐴 ⊆ On → (𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ On)) |
19 | 18 | ssrdv 3989 | . 2 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ⊆ On) |
20 | ordon 7764 | . . 3 ⊢ Ord On | |
21 | trssord 6382 | . . . 4 ⊢ ((Tr ∪ 𝐴 ∧ ∪ 𝐴 ⊆ On ∧ Ord On) → Ord ∪ 𝐴) | |
22 | 21 | 3exp 1120 | . . 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 397 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 ∪ cuni 4909 Tr wtr 5266 Ord word 6364 Oncon0 6365 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-ord 6368 df-on 6369 |
This theorem is referenced by: ssonuni 7767 ssonprc 7775 orduni 7777 onsucuni 7816 limuni3 7841 onfununi 8341 tfrlem8 8384 cofon1 8671 cofon2 8672 naddcllem 8675 onssnum 10035 unialeph 10096 cfslbn 10262 hsmexlem1 10421 inaprc 10831 bdayimaon 27196 noetasuplem4 27239 noetainflem4 27243 noeta2 27286 etasslt2 27315 scutbdaybnd2lim 27318 onsupneqmaxlim0 41973 onsupnmax 41977 onsupsucismax 42029 onsucunifi 42120 |
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