Theorem ranklim 4747

Description: The rank of a set belongs to a limit ordinal iff the rank of its power set does.

Assertion

Ref	Expression
ranklim	⊢ (Lim B → ((rank ‘A) ∈ B ↔ (rank ‘℘A) ∈ B))

Proof of Theorem ranklim

Step	Hyp	Ref	Expression
1		limsuc 3177	. . . 4 ⊢ (Lim B → ((rank ‘A) ∈ B ↔ suc (rank ‘A) ∈ B))
2	1	adantl 397	. . 3 ⊢ ((A ∈ V ⋀ Lim B) → ((rank ‘A) ∈ B ↔ suc (rank ‘A) ∈ B))
3		pweq 2455	. . . . . . . 8 ⊢ (x = A → ℘x = ℘A)
4	3	fveq2d 3785	. . . . . . 7 ⊢ (x = A → (rank ‘℘x) = (rank ‘℘A))
5		fveq2 3781	. . . . . . . 8 ⊢ (x = A → (rank ‘x) = (rank ‘A))
6		suceq 3091	. . . . . . . 8 ⊢ ((rank ‘x) = (rank ‘A) → suc (rank ‘x) = suc (rank ‘A))
7	5, 6	syl 10	. . . . . . 7 ⊢ (x = A → suc (rank ‘x) = suc (rank ‘A))
8	4, 7	eqeq12d 1536	. . . . . 6 ⊢ (x = A → ((rank ‘℘x) = suc (rank ‘x) ↔ (rank ‘℘A) = suc (rank ‘A)))
9		visset 1860	. . . . . . 7 ⊢ x ∈ V
10	9	rankpw 4746	. . . . . 6 ⊢ (rank ‘℘x) = suc (rank ‘x)
11	8, 10	vtoclg 1894	. . . . 5 ⊢ (A ∈ V → (rank ‘℘A) = suc (rank ‘A))
12	11	eleq1d 1587	. . . 4 ⊢ (A ∈ V → ((rank ‘℘A) ∈ B ↔ suc (rank ‘A) ∈ B))
13	12	adantr 398	. . 3 ⊢ ((A ∈ V ⋀ Lim B) → ((rank ‘℘A) ∈ B ↔ suc (rank ‘A) ∈ B))
14	2, 13	bitr4d 542	. 2 ⊢ ((A ∈ V ⋀ Lim B) → ((rank ‘A) ∈ B ↔ (rank ‘℘A) ∈ B))
15		fvprc 3778	. . . . 5 ⊢ (¬ A ∈ V → (rank ‘A) = ∅)
16		pwexb 2965	. . . . . . 7 ⊢ (A ∈ V ↔ ℘A ∈ V)
17	16	notbii 194	. . . . . 6 ⊢ (¬ A ∈ V ↔ ¬ ℘A ∈ V)
18		fvprc 3778	. . . . . 6 ⊢ (¬ ℘A ∈ V → (rank ‘℘A) = ∅)
19	17, 18	sylbi 206	. . . . 5 ⊢ (¬ A ∈ V → (rank ‘℘A) = ∅)
20	15, 19	eqtr4d 1557	. . . 4 ⊢ (¬ A ∈ V → (rank ‘A) = (rank ‘℘A))
21	20	eleq1d 1587	. . 3 ⊢ (¬ A ∈ V → ((rank ‘A) ∈ B ↔ (rank ‘℘A) ∈ B))
22	21	adantr 398	. 2 ⊢ ((¬ A ∈ V ⋀ Lim B) → ((rank ‘A) ∈ B ↔ (rank ‘℘A) ∈ B))
23	14, 22	pm2.61ian 487	1 ⊢ (Lim B → ((rank ‘A) ∈ B ↔ (rank ‘℘A) ∈ B))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 153   ⋀ wa 230   = wceq 997   ∈ wcel 999  Vcvv 1858  ∅c0 2331  ℘cpw 2453  Lim wlim 3006  suc csuc 3007   ‘cfv 3239  rankcrnk 4704

This theorem is referenced by:  rankxplim 4774

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922  ax-reg 4653  ax-inf2 4687

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-rab 1699  df-v 1859  df-sbc 1989  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-if 2414  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-int 2588  df-iun 2622  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-id 2891  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-lim 3010  df-suc 3011  df-om 3189  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-fv 3255  df-rdg 3990  df-r1 4705  df-rank 4706

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