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Mirrors > Home > MPE Home > Th. List > onuni | Structured version Visualization version GIF version |
Description: The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.) |
Ref | Expression |
---|---|
onuni | ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onss 7803 | . 2 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On) | |
2 | ssonuni 7798 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ⊆ On → ∪ 𝐴 ∈ On)) | |
3 | 1, 2 | mpd 15 | 1 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3962 ∪ cuni 4911 Oncon0 6385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 df-on 6389 |
This theorem is referenced by: onuninsuci 7860 oeeulem 8637 cnfcom3lem 9740 rankxpsuc 9919 dfac12lem2 10182 ttukeylem3 10548 r1limwun 10773 ontgval 36413 ordtoplem 36417 ordcmp 36429 1oequni2o 37350 rdgsucuni 37351 aomclem1 43042 omlimcl2 43230 onsucf1lem 43258 onsucf1olem 43259 onov0suclim 43263 dflim5 43318 |
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