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Theorem rankpwg 35636
Description: The rank of a power set. Closed form of rankpw 9834. (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 4608 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
21fveq2d 6885 . . 3 (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴))
3 fveq2 6881 . . . 4 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4 suceq 6420 . . . 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 2740 . 2 (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴)))
7 vex 3470 . . 3 𝑥 ∈ V
87rankpw 9834 . 2 (rank‘𝒫 𝑥) = suc (rank‘𝑥)
96, 8vtoclg 3535 1 (𝐴𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  𝒫 cpw 4594  suc csuc 6356  cfv 6533  rankcrnk 9754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-reg 9583  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-r1 9755  df-rank 9756
This theorem is referenced by:  hfpw  35652
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