| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 7805 | . 2 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On) | |
| 2 | ssonuni 7800 | . 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 2108 ⊆ wss 3951 ∪ cuni 4907 Oncon0 6384 |
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 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-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: onuninsuci 7861 oeeulem 8639 cnfcom3lem 9743 rankxpsuc 9922 dfac12lem2 10185 ttukeylem3 10551 r1limwun 10776 ontgval 36432 ordtoplem 36436 ordcmp 36448 1oequni2o 37369 rdgsucuni 37370 aomclem1 43066 omlimcl2 43254 onsucf1lem 43282 onsucf1olem 43283 onov0suclim 43287 dflim5 43342 |
| Copyright terms: Public domain | W3C validator |