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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 4081 | . 2 ⊢ (Ord B → Tr B) | |
2 | trss 3854 | . . . . 5 ⊢ (Tr B → (A ∈ B → A ⊆ B)) | |
3 | snssi 3499 | . . . . . 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 3111 | . . . 4 ⊢ ((A ⊆ B ∧ {A} ⊆ B) ↔ (A ∪ {A}) ⊆ B) | |
7 | 5, 6 | syl6ib 150 | . . 3 ⊢ (Tr B → (A ∈ B → (A ∪ {A}) ⊆ B)) |
8 | df-suc 4074 | . . . 4 ⊢ suc A = (A ∪ {A}) | |
9 | 8 | sseq1i 2963 | . . 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 1390 ∪ cun 2909 ⊆ wss 2911 {csn 3367 Tr wtr 3845 Ord word 4065 suc csuc 4068 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-uni 3572 df-tr 3846 df-iord 4069 df-suc 4074 |
This theorem is referenced by: ordelsuc 4197 tfrlemibfn 5883 sucinc2 5965 prarloclemn 6482 |
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