Theorem ordpwsuc 7824

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 3965	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On))
2		velpw 4611	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	2	anbi2ci 623	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))
4	1, 3	bitri 274	. . 3 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))
5		ordsssuc 6463	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴))
6	5	expcom 412	. . . . 5 ⊢ (Ord 𝐴 → (𝑥 ∈ On → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴)))
7	6	pm5.32d 575	. . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
8		simpr 483	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ suc 𝐴)
9		ordsuc 7822	. . . . . . 7 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
10		ordelon 6398	. . . . . . . 8 ⊢ ((Ord suc 𝐴 ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
11	10	ex 411	. . . . . . 7 ⊢ (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On))
12	9, 11	sylbi 216	. . . . . 6 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On))
13	12	ancrd 550	. . . . 5 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
14	8, 13	impbid2 225	. . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) ↔ 𝑥 ∈ suc 𝐴))
15	7, 14	bitrd 278	. . 3 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴) ↔ 𝑥 ∈ suc 𝐴))
16	4, 15	bitrid 282	. 2 ⊢ (Ord 𝐴 → (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ 𝑥 ∈ suc 𝐴))
17	16	eqrdv 2726	1 ⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ∩ cin 3948   ⊆ wss 3949  𝒫 cpw 4606  Ord word 6373  Oncon0 6374  suc csuc 6376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378  df-suc 6380

This theorem is referenced by:  onpwsuc  7825  orduniss2  7842