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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 7777, but does not use it directly, since ssorduni 7777 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 6374 | . . . 4 ⊢ (Ord 𝐵 → Tr 𝐵) | |
| 2 | 1 | ralimi 3100 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → ∀𝑥 ∈ 𝐴 Tr 𝐵) |
| 3 | triun 5232 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵) | |
| 4 | 2, 3 | syl 18 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵) |
| 5 | eliun 4959 | . . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
| 6 | nfra1 3287 | . . . . 5 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 Ord 𝐵 | |
| 7 | nfv 1942 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ On | |
| 8 | rsp 3251 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑥 ∈ 𝐴 → Ord 𝐵)) | |
| 9 | ordelon 6384 | . . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On) | |
| 10 | 9 | ex 417 | . . . . . 6 ⊢ (Ord 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On)) |
| 11 | 8, 10 | syl6 36 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))) |
| 12 | 6, 7, 11 | rexlimd 3270 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ On)) |
| 13 | 5, 12 | biimtrid 245 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ On)) |
| 14 | 13 | ssrdv 3942 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On) |
| 15 | ordon 7775 | . . 3 ⊢ Ord On | |
| 16 | trssord 6377 | . . . 4 ⊢ ((Tr ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On ∧ Ord On) → Ord ∪ 𝑥 ∈ 𝐴 𝐵) | |
| 17 | 16 | 3exp 1135 | . . 3 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On → (Ord On → Ord ∪ 𝑥 ∈ 𝐴 𝐵))) |
| 18 | 15, 17 | mpii 47 | . 2 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On → Ord ∪ 𝑥 ∈ 𝐴 𝐵)) |
| 19 | 4, 14, 18 | sylc 66 | 1 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2141 ∀wral 3077 ∃wrex 3087 ⊆ wss 3904 ∪ ciun 4955 Tr wtr 5217 Ord word 6359 Oncon0 6360 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-pr 5404 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3415 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-tr 5218 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-ord 6363 df-on 6364 |
| This theorem is referenced by: iunordi 50422 |
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