Theorem onpwsuc 7796

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 6356	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordpwsuc 7795	. 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 1560   ∈ wcel 2142   ∩ cin 3903  𝒫 cpw 4555  Ord word 6345  Oncon0 6346  suc csuc 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6349  df-on 6350  df-suc 6352

This theorem is referenced by: (None)