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Theorem ranklim 8922
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 7247 . . . 4 (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
21adantl 473 . . 3 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
3 pweq 4318 . . . . . . . 8 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
43fveq2d 6379 . . . . . . 7 (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴))
5 fveq2 6375 . . . . . . . 8 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
6 suceq 5973 . . . . . . . 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 2780 . . . . . 6 (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴)))
9 vex 3353 . . . . . . 7 𝑥 ∈ V
109rankpw 8921 . . . . . 6 (rank‘𝒫 𝑥) = suc (rank‘𝑥)
118, 10vtoclg 3418 . . . . 5 (𝐴 ∈ V → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
1211eleq1d 2829 . . . 4 (𝐴 ∈ V → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
1312adantr 472 . . 3 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
142, 13bitr4d 273 . 2 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
15 fvprc 6368 . . . . 5 𝐴 ∈ V → (rank‘𝐴) = ∅)
16 pwexb 7173 . . . . . 6 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
17 fvprc 6368 . . . . . 6 (¬ 𝒫 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
1816, 17sylnbi 321 . . . . 5 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
1915, 18eqtr4d 2802 . . . 4 𝐴 ∈ V → (rank‘𝐴) = (rank‘𝒫 𝐴))
2019eleq1d 2829 . . 3 𝐴 ∈ V → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2120adantr 472 . 2 ((¬ 𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2214, 21pm2.61ian 846 1 (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  c0 4079  𝒫 cpw 4315  Lim wlim 5909  suc csuc 5910  cfv 6068  rankcrnk 8841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-reg 8704  ax-inf2 8753
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-r1 8842  df-rank 8843
This theorem is referenced by:  rankxplim  8957
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