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Theorem ordpwsuc 6969
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 3779 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ On))
2 selpw 4142 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32anbi2ci 731 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
41, 3bitri 264 . . 3 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
5 ordsssuc 5776 . . . . . 6 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝑥 ∈ suc 𝐴))
65expcom 451 . . . . 5 (Ord 𝐴 → (𝑥 ∈ On → (𝑥𝐴𝑥 ∈ suc 𝐴)))
76pm5.32d 670 . . . 4 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
8 simpr 477 . . . . 5 ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ suc 𝐴)
9 ordsuc 6968 . . . . . . 7 (Ord 𝐴 ↔ Ord suc 𝐴)
10 ordelon 5711 . . . . . . . 8 ((Ord suc 𝐴𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
1110ex 450 . . . . . . 7 (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
129, 11sylbi 207 . . . . . 6 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
1312ancrd 576 . . . . 5 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴)))
148, 13impbid2 216 . . . 4 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴) ↔ 𝑥 ∈ suc 𝐴))
157, 14bitrd 268 . . 3 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ 𝑥 ∈ suc 𝐴))
164, 15syl5bb 272 . 2 (Ord 𝐴 → (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ 𝑥 ∈ suc 𝐴))
1716eqrdv 2619 1 (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cin 3558  wss 3559  𝒫 cpw 4135  Ord word 5686  Oncon0 5687  suc csuc 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691  df-suc 5693
This theorem is referenced by:  onpwsuc  6970  orduniss2  6987
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