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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 5733 | . 2 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
2 | ordsssuc 5812 | . 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) | |
3 | 1, 2 | sylan2 491 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 ⊆ wss 3574 Ord word 5722 Oncon0 5723 suc csuc 5725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-suc 5729 |
This theorem is referenced by: ordsssuc2 5814 onmindif 5815 tfindsg 7060 dfom2 7067 findsg 7093 ondif2 7582 oeeui 7682 cantnflem1 8586 rankr1bg 8666 rankr1c 8684 cofsmo 9091 cfsmolem 9092 cfcof 9096 fin1a2lem9 9230 alephreg 9404 winainflem 9515 onsuct0 32440 onint1 32448 |
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