Theorem ranklim 8707

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.

Step	Hyp	Ref	Expression
1		limsuc 7049	. . . 4 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
2	1	adantl 482	. . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
3		pweq 4161	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
4	3	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴))
5		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
6		suceq 5790	. . . . . . . 8 ⊢ ((rank‘𝑥) = (rank‘𝐴) → suc (rank‘𝑥) = suc (rank‘𝐴))
7	5, 6	syl 17	. . . . . . 7 ⊢ (𝑥 = 𝐴 → suc (rank‘𝑥) = suc (rank‘𝐴))
8	4, 7	eqeq12d 2637	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴)))
9		vex 3203	. . . . . . 7 ⊢ 𝑥 ∈ V
10	9	rankpw 8706	. . . . . 6 ⊢ (rank‘𝒫 𝑥) = suc (rank‘𝑥)
11	8, 10	vtoclg 3266	. . . . 5 ⊢ (𝐴 ∈ V → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
12	11	eleq1d 2686	. . . 4 ⊢ (𝐴 ∈ V → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
13	12	adantr 481	. . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
14	2, 13	bitr4d 271	. 2 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
15		fvprc 6185	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅)
16		pwexb 6975	. . . . . 6 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
17		fvprc 6185	. . . . . 6 ⊢ (¬ 𝒫 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
18	16, 17	sylnbi 320	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
19	15, 18	eqtr4d 2659	. . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = (rank‘𝒫 𝐴))
20	19	eleq1d 2686	. . 3 ⊢ (¬ 𝐴 ∈ V → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
21	20	adantr 481	. 2 ⊢ ((¬ 𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
22	14, 21	pm2.61ian 831	1 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  𝒫 cpw 4158  Lim wlim 5724  suc csuc 5725  ‘cfv 5888  rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  rankxplim  8742