Theorem onpwsuc 7525

Description: The collection of ordinal numbers in the power set of an ordinal number is its successor. (Contributed by NM, 19-Oct-2004.)

Assertion

Ref	Expression
onpwsuc	⊢ (𝐴 ∈ On → (𝒫 𝐴 ∩ On) = suc 𝐴)

Proof of Theorem onpwsuc

Step	Hyp	Ref	Expression
1		eloni 6195	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordpwsuc 7524	. 2 ⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ On → (𝒫 𝐴 ∩ On) = suc 𝐴)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110   ∩ cin 3934  𝒫 cpw 4538  Ord word 6184  Oncon0 6185  suc csuc 6187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-tr 5165  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-ord 6188  df-on 6189  df-suc 6191

This theorem is referenced by: (None)