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| Mirrors > Home > MPE Home > Th. List > limelon | Structured version Visualization version GIF version | ||
| Description: A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.) | 
| Ref | Expression | 
|---|---|
| limelon | ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | limord 6444 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
| 2 | elong 6392 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ On ↔ Ord 𝐴)) | |
| 3 | 1, 2 | imbitrrid 246 | . 2 ⊢ (𝐴 ∈ 𝐵 → (Lim 𝐴 → 𝐴 ∈ On)) | 
| 4 | 3 | imp 406 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Ord word 6383 Oncon0 6384 Lim wlim 6385 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-ss 3968 df-uni 4908 df-tr 5260 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-lim 6389 | 
| This theorem is referenced by: onzsl 7867 limuni3 7873 tfindsg2 7883 dfom2 7889 rdglim 8466 oalim 8570 omlim 8571 oelim 8572 oalimcl 8598 oaass 8599 omlimcl 8616 odi 8617 omass 8618 oen0 8624 oewordri 8630 oelim2 8633 oelimcl 8638 omabs 8689 r1lim 9812 alephordi 10114 cflm 10290 alephsing 10316 pwcfsdom 10623 winafp 10737 r1limwun 10776 omlimcl2 43254 oeord2lim 43322 | 
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