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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.
StepHypRef Expression
1 elin 3758 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ On))
2 selpw 4115 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32anbi2ci 728 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
41, 3bitri 263 . . 3 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
5 ordsssuc 5729 . . . . . 6 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝑥 ∈ suc 𝐴))
65expcom 450 . . . . 5 (Ord 𝐴 → (𝑥 ∈ On → (𝑥𝐴𝑥 ∈ suc 𝐴)))
76pm5.32d 669 . . . 4 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ (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 𝐴𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
1110ex 449 . . . . . . 7 (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
129, 11sylbi 206 . . . . . 6 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
1312ancrd 575 . . . . 5 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
148, 13impbid2 215 . . . 4 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) ↔ 𝑥 ∈ suc 𝐴))
157, 14bitrd 267 . . 3 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ 𝑥 ∈ suc 𝐴))
164, 15syl5bb 271 . 2 (Ord 𝐴 → (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ 𝑥 ∈ suc 𝐴))
1716eqrdv 2608 1 (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
 Colors of variables: wff setvar class 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
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