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Mirrors > Home > MPE Home > Th. List > onpwsuc | Structured version Visualization version GIF version |
Description: The collection of ordinal numbers in the power set of an ordinal number is its successor. (Contributed by NM, 19-Oct-2004.) |
Ref | Expression |
---|---|
onpwsuc | ⊢ (𝐴 ∈ On → (𝒫 𝐴 ∩ On) = suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6182 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordpwsuc 7549 | . 2 ⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ On → (𝒫 𝐴 ∩ On) = suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∩ cin 3842 𝒫 cpw 4488 Ord word 6171 Oncon0 6172 suc csuc 6174 |
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 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
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 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-ord 6175 df-on 6176 df-suc 6178 |
This theorem is referenced by: (None) |
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