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Theorem ranklim 9339
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 7577 . . . 4 (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
21adantl 485 . . 3 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
3 pweq 4501 . . . . . . . 8 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
43fveq2d 6672 . . . . . . 7 (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴))
5 fveq2 6668 . . . . . . . 8 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
6 suceq 6231 . . . . . . . 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 2754 . . . . . 6 (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴)))
9 vex 3401 . . . . . . 7 𝑥 ∈ V
109rankpw 9338 . . . . . 6 (rank‘𝒫 𝑥) = suc (rank‘𝑥)
118, 10vtoclg 3470 . . . . 5 (𝐴 ∈ V → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
1211eleq1d 2817 . . . 4 (𝐴 ∈ V → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
1312adantr 484 . . 3 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
142, 13bitr4d 285 . 2 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
15 fvprc 6660 . . . . 5 𝐴 ∈ V → (rank‘𝐴) = ∅)
16 pwexb 7501 . . . . . 6 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
17 fvprc 6660 . . . . . 6 (¬ 𝒫 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
1816, 17sylnbi 333 . . . . 5 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
1915, 18eqtr4d 2776 . . . 4 𝐴 ∈ V → (rank‘𝐴) = (rank‘𝒫 𝐴))
2019eleq1d 2817 . . 3 𝐴 ∈ V → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2120adantr 484 . 2 ((¬ 𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2214, 21pm2.61ian 812 1 (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  Vcvv 3397  c0 4209  𝒫 cpw 4485  Lim wlim 6167  suc csuc 6168  cfv 6333  rankcrnk 9258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-reg 9122  ax-inf2 9170
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-om 7594  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-r1 9259  df-rank 9260
This theorem is referenced by:  rankxplim  9374
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