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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 6394 | . 2 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
| 2 | ordsssuc 6473 | . 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) | |
| 3 | 1, 2 | sylan2 593 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 Ord word 6383 Oncon0 6384 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 |
| This theorem is referenced by: ordsssuc2 6475 onmindif 6476 tfindsg 7882 dfom2 7889 findsg 7919 ondif2 8540 oeeui 8640 cantnflem1 9729 rankr1bg 9843 rankr1c 9861 cofsmo 10309 cfsmolem 10310 cfcof 10314 fin1a2lem9 10448 alephreg 10622 winainflem 10733 n0sbday 28354 pw2bday 28418 zs12bday 28424 onsuct0 36442 onint1 36450 onintunirab 43239 cantnfresb 43337 oaun3lem4 43390 |
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