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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 6245 | . . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
2 | limeq 6197 | . . 3 ⊢ (𝐴 = ∪ 𝐴 → (Lim 𝐴 ↔ Lim ∪ 𝐴)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (Lim 𝐴 → (Lim 𝐴 ↔ Lim ∪ 𝐴)) |
4 | 3 | ibi 269 | 1 ⊢ (Lim 𝐴 → Lim ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∪ cuni 4831 Lim wlim 6186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-ne 3017 df-ral 3143 df-rex 3144 df-in 3942 df-ss 3951 df-uni 4832 df-tr 5165 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-ord 6188 df-lim 6190 |
This theorem is referenced by: rankxplim2 9303 rankxplim3 9304 |
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