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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 6263 | . 2 ⊢ Ord 𝐵 |
4 | ordelsuc 7515 | . 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) | |
5 | 1, 3, 4 | mp2an 691 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2111 ⊆ wss 3881 Ord word 6158 Oncon0 6159 suc csuc 6161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 df-suc 6165 |
This theorem is referenced by: omopthlem1 8265 rankval4 9280 rankc1 9283 rankc2 9284 rankxplim 9292 rankxplim3 9294 onsucsuccmpi 33904 |
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