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Mirrors > Home > ILE Home > Th. List > ordsucss | GIF version |
Description: The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.) |
Ref | Expression |
---|---|
ordsucss | ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 4115 | . 2 ⊢ (Ord 𝐵 → Tr 𝐵) | |
2 | trss 3863 | . . . . 5 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵)) | |
3 | snssi 3508 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)) |
5 | 2, 4 | jcad 291 | . . . 4 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → (𝐴 ⊆ 𝐵 ∧ {𝐴} ⊆ 𝐵))) |
6 | unss 3117 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ {𝐴} ⊆ 𝐵) ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵) | |
7 | 5, 6 | syl6ib 150 | . . 3 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → (𝐴 ∪ {𝐴}) ⊆ 𝐵)) |
8 | df-suc 4108 | . . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
9 | 8 | sseq1i 2969 | . . 3 ⊢ (suc 𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵) |
10 | 7, 9 | syl6ibr 151 | . 2 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
11 | 1, 10 | syl 14 | 1 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 ∪ cun 2915 ⊆ wss 2917 {csn 3375 Tr wtr 3854 Ord word 4099 suc csuc 4102 |
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-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-uni 3581 df-tr 3855 df-iord 4103 df-suc 4108 |
This theorem is referenced by: ordelsuc 4231 tfrlemibfn 5942 sucinc2 6026 nndomo 6326 prarloclemn 6597 |
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