Theorem ordpwsuc 7860

Description: The collection of ordinals in the power class of an ordinal is its successor. (Contributed by NM, 30-Jan-2005.)

Assertion

Ref	Expression
ordpwsuc	⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)

Proof of Theorem ordpwsuc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 4096	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On))
2		velpw 4565	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	2	anbi2ci 637	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))
4	1, 3	bitri 278	. . 3 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))
5		ordsssuc 6480	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴))
6	5	expcom 419	. . . . 5 ⊢ (Ord 𝐴 → (𝑥 ∈ On → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴)))
7	6	pm5.32d 588	. . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
8		simpr 490	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ suc 𝐴)
9		ordsuc 7859	. . . . . . 7 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
10		ordelon 6412	. . . . . . . 8 ⊢ ((Ord suc 𝐴 ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
11	10	ex 418	. . . . . . 7 ⊢ (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On))
12	9, 11	sylbi 220	. . . . . 6 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On))
13	12	ancrd 561	. . . . 5 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
14	8, 13	impbid2 229	. . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) ↔ 𝑥 ∈ suc 𝐴))
15	7, 14	bitrd 282	. . 3 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴) ↔ 𝑥 ∈ suc 𝐴))
16	4, 15	bitrid 286	. 2 ⊢ (Ord 𝐴 → (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ 𝑥 ∈ suc 𝐴))
17	16	eqrdv 2761	1 ⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  Ord word 6387  Oncon0 6388  suc csuc 6390

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392  df-suc 6394

This theorem is used by:  onpwsuc  7861  orduniss2  7878