Theorem rankpwg 9870

Description: The rank of the powerset is the successor of the rank. Closed form of rankpw 9869. (Contributed by Scott Fenton, 16-Jul-2015.)

Assertion

Ref	Expression
rankpwg	⊢ (𝐴 ∈ 𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))

Proof of Theorem rankpwg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4574	. . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2	1	fveq2d 6900	. . 3 ⊢ (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴))
3		fveq2 6896	. . . 4 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4		suceq 6438	. . . 4 ⊢ ((rank‘𝑥) = (rank‘𝐴) → suc (rank‘𝑥) = suc (rank‘𝐴))
5	3, 4	syl 18	. . 3 ⊢ (𝑥 = 𝐴 → suc (rank‘𝑥) = suc (rank‘𝐴))
6	2, 5	eqeq12d 2779	. 2 ⊢ (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴)))
7		vex 3458	. . 3 ⊢ 𝑥 ∈ V
8	7	rankpw 9869	. 2 ⊢ (rank‘𝒫 𝑥) = suc (rank‘𝑥)
9	6, 8	vtoclg 3521	1 ⊢ (𝐴 ∈ 𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  𝒫 cpw 4560  suc csuc 6371  ‘cfv 6547  rankcrnk 9780

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-reg 9599  ax-inf2 9655

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-om 7888  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-r1 9781  df-rank 9782

This theorem is used by:  hfpwOLD  9940