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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 6425 | . 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 2107 ⊆ wss 3950 Oncon0 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-v 3481 df-ss 3967 df-uni 4907 df-tr 5259 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 | 
| This theorem is referenced by: onelini 6501 oneluni 6502 oawordeulem 8593 cardsdomelir 10014 carddom2 10018 cardaleph 10130 alephsing 10317 domtriomlem 10483 axdc3lem 10491 inar1 10816 nodense 27738 | 
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