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Mirrors > Home > MPE Home > Th. List > onsucssi | Structured version Visualization version 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 |
---|---|
onssi.1 | ⊢ 𝐴 ∈ On |
onsucssi.2 | ⊢ 𝐵 ∈ On |
Ref | Expression |
---|---|
onsucssi | ⊢ (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onssi.1 | . 2 ⊢ 𝐴 ∈ On | |
2 | onsucssi.2 | . . 3 ⊢ 𝐵 ∈ On | |
3 | 2 | onordi 6466 | . 2 ⊢ Ord 𝐵 |
4 | ordelsuc 7802 | . 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) | |
5 | 1, 3, 4 | mp2an 689 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 ⊆ wss 3941 Ord word 6354 Oncon0 6355 suc csuc 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-tr 5257 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-ord 6358 df-on 6359 df-suc 6361 |
This theorem is referenced by: omopthlem1 8655 rankval4 9859 rankc1 9862 rankc2 9863 rankxplim 9871 rankxplim3 9873 cuteq1 27685 onsucsuccmpi 35819 |
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