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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 46081 | . 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 3076 | 1 ⊢ Ord ∪ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ∪ ciun 4918 Ord word 6229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 ax-un 7541 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-tr 5176 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-ord 6233 df-on 6234 |
This theorem is referenced by: (None) |
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