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Mirrors > Home > MPE Home > Th. List > onelssi | Structured version Visualization version GIF version |
Description: A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.) |
Ref | Expression |
---|---|
on.1 | ⊢ 𝐴 ∈ On |
Ref | Expression |
---|---|
onelssi | ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on.1 | . 2 ⊢ 𝐴 ∈ On | |
2 | onelss 6233 | . 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ⊆ wss 3936 Oncon0 6191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 df-in 3943 df-ss 3952 df-uni 4839 df-tr 5173 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 |
This theorem is referenced by: onelini 6302 oneluni 6303 oawordeulem 8180 cardsdomelir 9402 carddom2 9406 cardaleph 9515 alephsing 9698 domtriomlem 9864 axdc3lem 9872 inar1 10197 nodense 33196 |
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