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Theorem ranklim 8652
Description: The rank of a set belongs to a limit ordinal iff the rank of its power set does. (Contributed by NM, 18-Sep-2006.)
Assertion
Ref Expression
ranklim (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))

Proof of Theorem ranklim
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 limsuc 6997 . . . 4 (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
21adantl 482 . . 3 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
3 pweq 4138 . . . . . . . 8 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
43fveq2d 6154 . . . . . . 7 (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴))
5 fveq2 6150 . . . . . . . 8 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
6 suceq 5752 . . . . . . . 8 ((rank‘𝑥) = (rank‘𝐴) → suc (rank‘𝑥) = suc (rank‘𝐴))
75, 6syl 17 . . . . . . 7 (𝑥 = 𝐴 → suc (rank‘𝑥) = suc (rank‘𝐴))
84, 7eqeq12d 2641 . . . . . 6 (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴)))
9 vex 3194 . . . . . . 7 𝑥 ∈ V
109rankpw 8651 . . . . . 6 (rank‘𝒫 𝑥) = suc (rank‘𝑥)
118, 10vtoclg 3257 . . . . 5 (𝐴 ∈ V → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
1211eleq1d 2688 . . . 4 (𝐴 ∈ V → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
1312adantr 481 . . 3 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
142, 13bitr4d 271 . 2 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
15 fvprc 6144 . . . . 5 𝐴 ∈ V → (rank‘𝐴) = ∅)
16 pwexb 6923 . . . . . 6 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
17 fvprc 6144 . . . . . 6 (¬ 𝒫 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
1816, 17sylnbi 320 . . . . 5 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
1915, 18eqtr4d 2663 . . . 4 𝐴 ∈ V → (rank‘𝐴) = (rank‘𝒫 𝐴))
2019eleq1d 2688 . . 3 𝐴 ∈ V → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2120adantr 481 . 2 ((¬ 𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2214, 21pm2.61ian 830 1 (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  Vcvv 3191  c0 3896  𝒫 cpw 4135  Lim wlim 5686  suc csuc 5687  cfv 5850  rankcrnk 8571
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-reg 8442  ax-inf2 8483
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  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-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-r1 8572  df-rank 8573
This theorem is referenced by:  rankxplim  8687
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