Theorem ordpwsuc 6907

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 3758	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On))
2		selpw 4115	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	2	anbi2ci 728	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))
4	1, 3	bitri 263	. . 3 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))
5		ordsssuc 5729	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴))
6	5	expcom 450	. . . . 5 ⊢ (Ord 𝐴 → (𝑥 ∈ On → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴)))
7	6	pm5.32d 669	. . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
8		simpr 476	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ suc 𝐴)
9		ordsuc 6906	. . . . . . 7 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
10		ordelon 5664	. . . . . . . 8 ⊢ ((Ord suc 𝐴 ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
11	10	ex 449	. . . . . . 7 ⊢ (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On))
12	9, 11	sylbi 206	. . . . . 6 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On))
13	12	ancrd 575	. . . . 5 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
14	8, 13	impbid2 215	. . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) ↔ 𝑥 ∈ suc 𝐴))
15	7, 14	bitrd 267	. . 3 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴) ↔ 𝑥 ∈ suc 𝐴))
16	4, 15	syl5bb 271	. 2 ⊢ (Ord 𝐴 → (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ 𝑥 ∈ suc 𝐴))
17	16	eqrdv 2608	1 ⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  Ord word 5639  Oncon0 5640  suc csuc 5642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646

This theorem is referenced by:  onpwsuc  6908  orduniss2  6925