Theorem r1pw 9276

Description: A stronger property of 𝑅₁ than rankpw 9274. The latter merely proves that 𝑅₁ of the successor is a power set, but here we prove that if 𝐴 is in the cumulative hierarchy, then 𝒫 𝐴 is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

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

Proof of Theorem r1pw

Step	Hyp	Ref	Expression
1		rankpwi 9254	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
2	1	eleq1d 2899	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
3		eloni 6203	. . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordsucelsuc 7539	. . . . . . 7 ⊢ (Ord 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
5	3, 4	syl 17	. . . . . 6 ⊢ (𝐵 ∈ On → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
6	5	bicomd 225	. . . . 5 ⊢ (𝐵 ∈ On → (suc (rank‘𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
7	2, 6	sylan9bb 512	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
8		pwwf 9238	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
9	8	biimpi 218	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
10		suceloni 7530	. . . . . 6 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
11		r1fnon 9198	. . . . . . 7 ⊢ 𝑅1 Fn On
12		fndm 6457	. . . . . . 7 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
13	11, 12	ax-mp 5	. . . . . 6 ⊢ dom 𝑅1 = On
14	10, 13	eleqtrrdi 2926	. . . . 5 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ dom 𝑅1)
15		rankr1ag 9233	. . . . 5 ⊢ ((𝒫 𝐴 ∈ ∪ (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
16	9, 14, 15	syl2an 597	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
17	13	eleq2i 2906	. . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On)
18		rankr1ag 9233	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
19	17, 18	sylan2br 596	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
20	7, 16, 19	3bitr4rd 314	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
21	20	ex 415	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
22		r1elwf 9227	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
23		r1elwf 9227	. . . . . 6 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
24		r1elssi 9236	. . . . . 6 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ ∪ (𝑅1 “ On))
25	23, 24	syl 17	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 ⊆ ∪ (𝑅1 “ On))
26		ssid 3991	. . . . . 6 ⊢ 𝐴 ⊆ 𝐴
27		pwexr 7489	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ V)
28		elpwg 4544	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝐴))
29	27, 28	syl 17	. . . . . 6 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → (𝐴 ∈ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝐴))
30	26, 29	mpbiri 260	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ 𝒫 𝐴)
31	25, 30	sseldd 3970	. . . 4 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
32	22, 31	pm5.21ni 381	. . 3 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
33	32	a1d 25	. 2 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
34	21, 33	pm2.61i 184	1 ⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ⊆ wss 3938  𝒫 cpw 4541  ∪ cuni 4840  dom cdm 5557   “ cima 5560  Ord word 6192  Oncon0 6193  suc csuc 6195   Fn wfn 6352  ‘cfv 6357  𝑅₁cr1 9193  rankcrnk 9194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-r1 9195  df-rank 9196

This theorem is referenced by:  inatsk  10202