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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iunordi | Structured version Visualization version GIF version | ||
| Description: The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. (Contributed by Emmett Weisz, 3-Nov-2019.) |
| Ref | Expression |
|---|---|
| iunordi.B | ⊢ Ord 𝐵 |
| Ref | Expression |
|---|---|
| iunordi | ⊢ Ord ∪ 𝑥 ∈ 𝐴 𝐵 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iunord 49195 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵) | |
| 2 | iunordi.B | . . 3 ⊢ Ord 𝐵 | |
| 3 | 2 | a1i 11 | . 2 ⊢ (𝑥 ∈ 𝐴 → Ord 𝐵) |
| 4 | 1, 3 | mprg 3067 | 1 ⊢ Ord ∪ 𝑥 ∈ 𝐴 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ∪ ciun 4991 Ord word 6383 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: (None) |
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