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Mirrors > Home > ILE Home > Th. List > limuni2 | 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 4151 | . . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
2 | limeq 4132 | . . 3 ⊢ (𝐴 = ∪ 𝐴 → (Lim 𝐴 ↔ Lim ∪ 𝐴)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (Lim 𝐴 → (Lim 𝐴 ↔ Lim ∪ 𝐴)) |
4 | 3 | ibi 174 | 1 ⊢ (Lim 𝐴 → Lim ∪ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∪ cuni 3601 Lim wlim 4119 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-in 2979 df-ss 2986 df-uni 3602 df-tr 3876 df-iord 4121 df-ilim 4124 |
This theorem is referenced by: (None) |
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