Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6278 | . 2 ⊢ Ord 𝐵 |
4 | ordelsuc 7557 | . 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) | |
5 | 1, 3, 4 | mp2an 692 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2114 ⊆ wss 3844 Ord word 6172 Oncon0 6173 suc csuc 6175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-11 2162 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-tr 5138 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-ord 6176 df-on 6177 df-suc 6179 |
This theorem is referenced by: omopthlem1 8316 rankval4 9372 rankc1 9375 rankc2 9376 rankxplim 9384 rankxplim3 9386 onsucsuccmpi 34278 |
Copyright terms: Public domain | W3C validator |