Theorem r1pw 9769

Description: A stronger property of 𝑅₁ than rankpw 9767. 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 9747	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
2	1	eleq1d 2821	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
3		eloni 6333	. . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordsucelsuc 7773	. . . . . . 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 223	. . . . 5 ⊢ (𝐵 ∈ On → (suc (rank‘𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
7	2, 6	sylan9bb 509	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
8		pwwf 9731	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
9	8	biimpi 216	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
10		onsuc 7764	. . . . . 6 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
11		r1fnon 9691	. . . . . . 7 ⊢ 𝑅1 Fn On
12	11	fndmi 6602	. . . . . 6 ⊢ dom 𝑅1 = On
13	10, 12	eleqtrrdi 2847	. . . . 5 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ dom 𝑅1)
14		rankr1ag 9726	. . . . 5 ⊢ ((𝒫 𝐴 ∈ ∪ (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
15	9, 13, 14	syl2an 597	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
16	12	eleq2i 2828	. . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On)
17		rankr1ag 9726	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
18	16, 17	sylan2br 596	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
19	7, 15, 18	3bitr4rd 312	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
20	19	ex 412	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
21		r1elwf 9720	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
22		r1elwf 9720	. . . . . 6 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
23		r1elssi 9729	. . . . . 6 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ ∪ (𝑅1 “ On))
24	22, 23	syl 17	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 ⊆ ∪ (𝑅1 “ On))
25		ssid 3944	. . . . . 6 ⊢ 𝐴 ⊆ 𝐴
26		pwexr 7719	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ V)
27		elpwg 4544	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝐴))
28	26, 27	syl 17	. . . . . 6 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → (𝐴 ∈ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝐴))
29	25, 28	mpbiri 258	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ 𝒫 𝐴)
30	24, 29	sseldd 3922	. . . 4 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
31	21, 30	pm5.21ni 377	. . 3 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
32	31	a1d 25	. 2 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
33	20, 32	pm2.61i 182	1 ⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  Vcvv 3429   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850  dom cdm 5631   “ cima 5634  Ord word 6322  Oncon0 6323  suc csuc 6325  ‘cfv 6498  𝑅₁cr1 9686  rankcrnk 9687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-r1 9688  df-rank 9689

This theorem is referenced by:  inatsk  10701