MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordpwsuc Structured version   Visualization version   GIF version

Theorem ordpwsuc 7835
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 3979 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ On))
2 velpw 4610 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32anbi2ci 625 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
41, 3bitri 275 . . 3 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
5 ordsssuc 6475 . . . . . 6 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝑥 ∈ suc 𝐴))
65expcom 413 . . . . 5 (Ord 𝐴 → (𝑥 ∈ On → (𝑥𝐴𝑥 ∈ suc 𝐴)))
76pm5.32d 577 . . . 4 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
8 simpr 484 . . . . 5 ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ suc 𝐴)
9 ordsuc 7833 . . . . . . 7 (Ord 𝐴 ↔ Ord suc 𝐴)
10 ordelon 6410 . . . . . . . 8 ((Ord suc 𝐴𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
1110ex 412 . . . . . . 7 (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
129, 11sylbi 217 . . . . . 6 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
1312ancrd 551 . . . . 5 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
148, 13impbid2 226 . . . 4 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) ↔ 𝑥 ∈ suc 𝐴))
157, 14bitrd 279 . . 3 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ 𝑥 ∈ suc 𝐴))
164, 15bitrid 283 . 2 (Ord 𝐴 → (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ 𝑥 ∈ suc 𝐴))
1716eqrdv 2733 1 (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cin 3962  wss 3963  𝒫 cpw 4605  Ord word 6385  Oncon0 6386  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by:  onpwsuc  7836  orduniss2  7853
  Copyright terms: Public domain W3C validator