Theorem r1pw 3529

Description: A stronger property of R₁ than rankpw 3528. The latter merely proves that R₁ of the successor is a powerset, but here we prove that if A is in the cumulative hierarchy, then ℘A is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-04.)

Assertion

Ref	Expression
r1pw	⊢ (B ∈ On → (A ∈ (R1 ‘B) ↔ ℘A ∈ (R1 ‘suc B)))

Proof of Theorem r1pw

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 5 ⊢ (x = A → (x ∈ (R1 ‘B) ↔ A ∈ (R1 ‘B)))
2		pweq 1800	. . . . . 6 ⊢ (x = A → ℘x = ℘A)
3	2	eleq1d 1155	. . . . 5 ⊢ (x = A → (℘x ∈ (R1 ‘suc B) ↔ ℘A ∈ (R1 ‘suc B)))
4	1, 3	bibi12d 477	. . . 4 ⊢ (x = A → ((x ∈ (R1 ‘B) ↔ ℘x ∈ (R1 ‘suc B)) ↔ (A ∈ (R1 ‘B) ↔ ℘A ∈ (R1 ‘suc B))))
5	4	imbi2d 464	. . 3 ⊢ (x = A → ((B ∈ On → (x ∈ (R1 ‘B) ↔ ℘x ∈ (R1 ‘suc B))) ↔ (B ∈ On → (A ∈ (R1 ‘B) ↔ ℘A ∈ (R1 ‘suc B)))))
6		visset 1350	. . . . . . 7 ⊢ x ∈ V
7	6	rankr1a 3521	. . . . . 6 ⊢ (B ∈ On → (x ∈ (R1 ‘B) ↔ (rank ‘x) ∈ B))
8		eloni 2209	. . . . . . 7 ⊢ (B ∈ On → Ord B)
9		ordsucelsuc 2324	. . . . . . 7 ⊢ (Ord B → ((rank ‘x) ∈ B ↔ suc (rank ‘x) ∈ suc B))
10	8, 9	syl 12	. . . . . 6 ⊢ (B ∈ On → ((rank ‘x) ∈ B ↔ suc (rank ‘x) ∈ suc B))
11	7, 10	bitrd 406	. . . . 5 ⊢ (B ∈ On → (x ∈ (R1 ‘B) ↔ suc (rank ‘x) ∈ suc B))
12	6	rankpw 3528	. . . . . 6 ⊢ (rank ‘℘x) = suc (rank ‘x)
13	12	eleq1i 1152	. . . . 5 ⊢ ((rank ‘℘x) ∈ suc B ↔ suc (rank ‘x) ∈ suc B)
14	11, 13	syl6bbr 416	. . . 4 ⊢ (B ∈ On → (x ∈ (R1 ‘B) ↔ (rank ‘℘x) ∈ suc B))
15		suceloni 2314	. . . . 5 ⊢ (B ∈ On → suc B ∈ On)
16	6	pwex 1806	. . . . . 6 ⊢ ℘x ∈ V
17	16	rankr1a 3521	. . . . 5 ⊢ (suc B ∈ On → (℘x ∈ (R1 ‘suc B) ↔ (rank ‘℘x) ∈ suc B))
18	15, 17	syl 12	. . . 4 ⊢ (B ∈ On → (℘x ∈ (R1 ‘suc B) ↔ (rank ‘℘x) ∈ suc B))
19	14, 18	bitr4d 409	. . 3 ⊢ (B ∈ On → (x ∈ (R1 ‘B) ↔ ℘x ∈ (R1 ‘suc B)))
20	5, 19	vtoclg 1383	. 2 ⊢ (A ∈ V → (B ∈ On → (A ∈ (R1 ‘B) ↔ ℘A ∈ (R1 ‘suc B))))
21		elisset 1354	. . . 4 ⊢ (A ∈ (R1 ‘B) → A ∈ V)
22		elisset 1354	. . . . 5 ⊢ (℘A ∈ (R1 ‘suc B) → ℘A ∈ V)
23		pwexb 1963	. . . . 5 ⊢ (A ∈ V ↔ ℘A ∈ V)
24	22, 23	sylibr 175	. . . 4 ⊢ (℘A ∈ (R1 ‘suc B) → A ∈ V)
25	21, 24	pm5.21ni 503	. . 3 ⊢ (¬ A ∈ V → (A ∈ (R1 ‘B) ↔ ℘A ∈ (R1 ‘suc B)))
26	25	a1d 14	. 2 ⊢ (¬ A ∈ V → (B ∈ On → (A ∈ (R1 ‘B) ↔ ℘A ∈ (R1 ‘suc B))))
27	20, 26	pm2.61i 110	1 ⊢ (B ∈ On → (A ∈ (R1 ‘B) ↔ ℘A ∈ (R1 ‘suc B)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ℘cpw 1798  Ord word 2198  Oncon0 2199  suc csuc 2201   ‘cfv 2422  R₁cr1 3485  rankcrnk 3486

This theorem is referenced by:  r1pwcl 3530

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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