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Theorem rankpwg 36148
Description: The rank of a power set. Closed form of rankpw 9879. (Contributed by Scott Fenton, 16-Jul-2015.)
Assertion
Ref Expression
rankpwg (𝐴𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))

Proof of Theorem rankpwg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pweq 4612 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
21fveq2d 6908 . . 3 (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴))
3 fveq2 6904 . . . 4 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4 suceq 6448 . . . 4 ((rank‘𝑥) = (rank‘𝐴) → suc (rank‘𝑥) = suc (rank‘𝐴))
53, 4syl 17 . . 3 (𝑥 = 𝐴 → suc (rank‘𝑥) = suc (rank‘𝐴))
62, 5eqeq12d 2752 . 2 (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴)))
7 vex 3483 . . 3 𝑥 ∈ V
87rankpw 9879 . 2 (rank‘𝒫 𝑥) = suc (rank‘𝑥)
96, 8vtoclg 3553 1 (𝐴𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  𝒫 cpw 4598  suc csuc 6384  cfv 6559  rankcrnk 9799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751  ax-reg 9628  ax-inf2 9677
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-int 4945  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-om 7884  df-2nd 8011  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-r1 9800  df-rank 9801
This theorem is referenced by:  hfpw  36164
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