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Mathbox for Emmett Weisz |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunord | Structured version Visualization version GIF version |
Description: The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7762, but does not use it directly, since ssorduni 7762 does not work when 𝐵 is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.) |
Ref | Expression |
---|---|
iunord | ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 6371 | . . . 4 ⊢ (Ord 𝐵 → Tr 𝐵) | |
2 | 1 | ralimi 3077 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → ∀𝑥 ∈ 𝐴 Tr 𝐵) |
3 | triun 5273 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵) |
5 | eliun 4994 | . . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
6 | nfra1 3275 | . . . . 5 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 Ord 𝐵 | |
7 | nfv 1909 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ On | |
8 | rsp 3238 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑥 ∈ 𝐴 → Ord 𝐵)) | |
9 | ordelon 6381 | . . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On) | |
10 | 9 | ex 412 | . . . . . 6 ⊢ (Ord 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On)) |
11 | 8, 10 | syl6 35 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))) |
12 | 6, 7, 11 | rexlimd 3257 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ On)) |
13 | 5, 12 | biimtrid 241 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ On)) |
14 | 13 | ssrdv 3983 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On) |
15 | ordon 7760 | . . 3 ⊢ Ord On | |
16 | trssord 6374 | . . . 4 ⊢ ((Tr ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On ∧ Ord On) → Ord ∪ 𝑥 ∈ 𝐴 𝐵) | |
17 | 16 | 3exp 1116 | . . 3 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On → (Ord On → Ord ∪ 𝑥 ∈ 𝐴 𝐵))) |
18 | 15, 17 | mpii 46 | . 2 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On → Ord ∪ 𝑥 ∈ 𝐴 𝐵)) |
19 | 4, 14, 18 | sylc 65 | 1 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 ⊆ wss 3943 ∪ ciun 4990 Tr wtr 5258 Ord word 6356 Oncon0 6357 |
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 |
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-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 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-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-ord 6360 df-on 6361 |
This theorem is referenced by: iunordi 47977 |
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