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Mirrors > Home > MPE Home > Th. List > limuni2 | Structured version Visualization version GIF version |
Description: The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.) |
Ref | Expression |
---|---|
limuni2 | ⊢ (Lim 𝐴 → Lim ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limuni 6290 | . . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
2 | limeq 6242 | . . 3 ⊢ (𝐴 = ∪ 𝐴 → (Lim 𝐴 ↔ Lim ∪ 𝐴)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (Lim 𝐴 → (Lim 𝐴 ↔ Lim ∪ 𝐴)) |
4 | 3 | ibi 270 | 1 ⊢ (Lim 𝐴 → Lim ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∪ cuni 4833 Lim wlim 6231 |
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-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1091 df-tru 1546 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ne 2942 df-ral 3067 df-v 3422 df-in 3887 df-ss 3897 df-uni 4834 df-tr 5176 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-ord 6233 df-lim 6235 |
This theorem is referenced by: rankxplim2 9520 rankxplim3 9521 |
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