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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 5650 | . 2 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
2 | ordsssuc 5729 | . 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) | |
3 | 1, 2 | sylan2 490 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 Ord word 5639 Oncon0 5640 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-suc 5646 |
This theorem is referenced by: ordsssuc2 5731 onmindif 5732 tfindsg 6952 dfom2 6959 findsg 6985 ondif2 7469 oeeui 7569 cantnflem1 8469 rankr1bg 8549 rankr1c 8567 cofsmo 8974 cfsmolem 8975 cfcof 8979 fin1a2lem9 9113 alephreg 9283 winainflem 9394 nobndlem8 31098 onsuct0 31610 onint1 31618 |
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