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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 4552 | . . 3 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) | |
| 2 | 1 | adantl 277 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
| 3 | sucssel 4471 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵)) | |
| 4 | 3 | adantr 276 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵)) |
| 5 | 2, 4 | impbid 129 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2176 ⊆ wss 3166 Ord word 4409 suc csuc 4412 |
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 |
| This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-sn 3639 df-uni 3851 df-tr 4143 df-iord 4413 df-suc 4418 |
| This theorem is referenced by: onsucssi 4554 onsucmin 4555 onsucelsucr 4556 onsucsssucr 4557 onsucsssucexmid 4575 frecsuclem 6492 ordgt0ge1 6521 nnsucsssuc 6578 ennnfonelemk 12771 nninfsellemeq 15951 |
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