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Mirrors > Home > MPE Home > Th. List > onsssuc | Structured version Visualization version GIF version |
Description: A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.) |
Ref | Expression |
---|---|
onsssuc | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6201 | . 2 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
2 | ordsssuc 6277 | . 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) | |
3 | 1, 2 | sylan2 596 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2112 ⊆ wss 3853 Ord word 6190 Oncon0 6191 suc csuc 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 df-on 6195 df-suc 6197 |
This theorem is referenced by: ordsssuc2 6279 onmindif 6280 tfindsg 7617 dfom2 7624 findsg 7655 ondif2 8207 oeeui 8308 cantnflem1 9282 rankr1bg 9384 rankr1c 9402 cofsmo 9848 cfsmolem 9849 cfcof 9853 fin1a2lem9 9987 alephreg 10161 winainflem 10272 onsuct0 34316 onint1 34324 |
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