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Mirrors > Home > ILE Home > Th. List > ordelsuc | GIF version |
Description: A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.) |
Ref | Expression |
---|---|
ordelsuc | ⊢ ((A ∈ 𝐶 ∧ Ord B) → (A ∈ B ↔ suc A ⊆ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsucss 4196 | . . 3 ⊢ (Ord B → (A ∈ B → suc A ⊆ B)) | |
2 | 1 | adantl 262 | . 2 ⊢ ((A ∈ 𝐶 ∧ Ord B) → (A ∈ B → suc A ⊆ B)) |
3 | sucssel 4127 | . . 3 ⊢ (A ∈ 𝐶 → (suc A ⊆ B → A ∈ B)) | |
4 | 3 | adantr 261 | . 2 ⊢ ((A ∈ 𝐶 ∧ Ord B) → (suc A ⊆ B → A ∈ B)) |
5 | 2, 4 | impbid 120 | 1 ⊢ ((A ∈ 𝐶 ∧ Ord B) → (A ∈ B ↔ suc A ⊆ B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1390 ⊆ wss 2911 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: onsucmin 4198 onsucelsucr 4199 onsucsssucr 4200 onsucsssucexmid 4212 ordgt0ge1 5957 nnsucsssuc 6010 |
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