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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 5701 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
2 | elong 5648 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ On ↔ Ord 𝐴)) | |
3 | 1, 2 | syl5ibr 235 | . 2 ⊢ (𝐴 ∈ 𝐵 → (Lim 𝐴 → 𝐴 ∈ On)) |
4 | 3 | imp 444 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 Ord word 5639 Oncon0 5640 Lim wlim 5641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-uni 4373 df-tr 4681 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 |
This theorem is referenced by: onzsl 6938 limuni3 6944 tfindsg2 6953 dfom2 6959 rdglim 7409 oalim 7499 omlim 7500 oelim 7501 oalimcl 7527 oaass 7528 omlimcl 7545 odi 7546 omass 7547 oen0 7553 oewordri 7559 oelim2 7562 oelimcl 7567 omabs 7614 r1lim 8518 alephordi 8780 cflm 8955 alephsing 8981 pwcfsdom 9284 winafp 9398 r1limwun 9437 |
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