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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 6227 | . 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 2110 ⊆ wss 3935 Oncon0 6185 |
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-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3496 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-on 6189 |
This theorem is referenced by: onelini 6296 oneluni 6297 oawordeulem 8174 cardsdomelir 9396 carddom2 9400 cardaleph 9509 alephsing 9692 domtriomlem 9858 axdc3lem 9866 inar1 10191 nodense 33191 |
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