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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 4438 | . 2 ⊢ Ord 𝐵 |
4 | ordelsuc 4516 | . 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 2158 ⊆ wss 3141 Ord word 4374 Oncon0 4375 suc csuc 4377 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-uni 3822 df-tr 4114 df-iord 4378 df-on 4380 df-suc 4383 |
This theorem is referenced by: (None) |
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