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Theorem ordpwsuc 7535
 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 3876 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ On))
2 velpw 4502 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32anbi2ci 627 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
41, 3bitri 278 . . 3 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
5 ordsssuc 6260 . . . . . 6 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝑥 ∈ suc 𝐴))
65expcom 417 . . . . 5 (Ord 𝐴 → (𝑥 ∈ On → (𝑥𝐴𝑥 ∈ suc 𝐴)))
76pm5.32d 580 . . . 4 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
8 simpr 488 . . . . 5 ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ suc 𝐴)
9 ordsuc 7534 . . . . . . 7 (Ord 𝐴 ↔ Ord suc 𝐴)
10 ordelon 6198 . . . . . . . 8 ((Ord suc 𝐴𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
1110ex 416 . . . . . . 7 (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
129, 11sylbi 220 . . . . . 6 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
1312ancrd 555 . . . . 5 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
148, 13impbid2 229 . . . 4 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) ↔ 𝑥 ∈ suc 𝐴))
157, 14bitrd 282 . . 3 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ 𝑥 ∈ suc 𝐴))
164, 15syl5bb 286 . 2 (Ord 𝐴 → (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ 𝑥 ∈ suc 𝐴))
1716eqrdv 2756 1 (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∩ cin 3859   ⊆ wss 3860  𝒫 cpw 4497  Ord word 6173  Oncon0 6174  suc csuc 6176 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-tr 5143  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-ord 6177  df-on 6178  df-suc 6180 This theorem is referenced by:  onpwsuc  7536  orduniss2  7553
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