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Mirrors > Home > ILE Home > Th. List > onsucssi | GIF version |
Description: A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.) |
Ref | Expression |
---|---|
onsucssi.1 | ⊢ 𝐴 ∈ On |
onsucssi.2 | ⊢ 𝐵 ∈ On |
Ref | Expression |
---|---|
onsucssi | ⊢ (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsucssi.1 | . 2 ⊢ 𝐴 ∈ On | |
2 | onsucssi.2 | . . 3 ⊢ 𝐵 ∈ On | |
3 | 2 | onordi 4441 | . 2 ⊢ Ord 𝐵 |
4 | ordelsuc 4519 | . 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) | |
5 | 1, 3, 4 | mp2an 426 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2160 ⊆ wss 3144 Ord word 4377 Oncon0 4378 suc csuc 4380 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-uni 3825 df-tr 4117 df-iord 4381 df-on 4383 df-suc 4386 |
This theorem is referenced by: (None) |
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