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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 5683 | . 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 1977 ⊆ wss 3540 Oncon0 5640 |
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-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 |
This theorem is referenced by: onelini 5756 oneluni 5757 oawordeulem 7521 cardsdomelir 8682 carddom2 8686 cardaleph 8795 alephsing 8981 domtriomlem 9147 axdc3lem 9155 inar1 9476 nodenselem6 31085 nodense 31088 |
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