Theorem onpwsuc 7550

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 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)

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)