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Mirrors > Home > ILE Home > Th. List > ordelss | GIF version |
Description: An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.) |
Ref | Expression |
---|---|
ordelss | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 4115 | . 2 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | trss 3863 | . . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | |
3 | 2 | imp 115 | . 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴) |
4 | 1, 3 | sylan 267 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 ⊆ wss 2917 Tr wtr 3854 Ord word 4099 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-in 2924 df-ss 2931 df-uni 3581 df-tr 3855 df-iord 4103 |
This theorem is referenced by: ordelord 4118 onelss 4124 ordsuc 4287 smores3 5908 tfrlem1 5923 tfrlemisucaccv 5939 tfrlemiubacc 5944 nntri1 6074 nnsseleq 6079 ordiso2 6357 |
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