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Mirrors > Home > ILE Home > Th. List > onelssi | 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 4387 | . 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ⊆ wss 3129 Oncon0 4363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-in 3135 df-ss 3142 df-uni 3810 df-tr 4102 df-iord 4366 df-on 4368 |
This theorem is referenced by: onelini 4430 oneluni 4431 omp1eomlem 7092 enumctlemm 7112 ennnfonelemdc 12394 ctinfom 12423 2o01f 14666 isomninnlem 14698 iswomninnlem 14717 ismkvnnlem 14720 |
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