Theorem ordpwsuc 7851

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 3992	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On))
2		velpw 4627	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	2	anbi2ci 624	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))
4	1, 3	bitri 275	. . 3 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))
5		ordsssuc 6484	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴))
6	5	expcom 413	. . . . 5 ⊢ (Ord 𝐴 → (𝑥 ∈ On → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴)))
7	6	pm5.32d 576	. . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
8		simpr 484	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ suc 𝐴)
9		ordsuc 7849	. . . . . . 7 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
10		ordelon 6419	. . . . . . . 8 ⊢ ((Ord suc 𝐴 ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
11	10	ex 412	. . . . . . 7 ⊢ (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On))
12	9, 11	sylbi 217	. . . . . 6 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On))
13	12	ancrd 551	. . . . 5 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
14	8, 13	impbid2 226	. . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) ↔ 𝑥 ∈ suc 𝐴))
15	7, 14	bitrd 279	. . 3 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴) ↔ 𝑥 ∈ suc 𝐴))
16	4, 15	bitrid 283	. 2 ⊢ (Ord 𝐴 → (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ 𝑥 ∈ suc 𝐴))
17	16	eqrdv 2738	1 ⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  Ord word 6394  Oncon0 6395  suc csuc 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-suc 6401

This theorem is referenced by:  onpwsuc  7852  orduniss2  7869