Theorem r1pw 9744

Description: A stronger property of 𝑅₁ than rankpw 9742. 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 9722	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
2	1	eleq1d 2816	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
3		eloni 6322	. . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordsucelsuc 7758	. . . . . . 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 9706	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
9	8	biimpi 216	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
10		onsuc 7749	. . . . . 6 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
11		r1fnon 9666	. . . . . . 7 ⊢ 𝑅1 Fn On
12	11	fndmi 6591	. . . . . 6 ⊢ dom 𝑅1 = On
13	10, 12	eleqtrrdi 2842	. . . . 5 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ dom 𝑅1)
14		rankr1ag 9701	. . . . 5 ⊢ ((𝒫 𝐴 ∈ ∪ (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
15	9, 13, 14	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
16	12	eleq2i 2823	. . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On)
17		rankr1ag 9701	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
18	16, 17	sylan2br 595	. . . 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 9695	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
22		r1elwf 9695	. . . . . 6 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
23		r1elssi 9704	. . . . . 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 3952	. . . . . 6 ⊢ 𝐴 ⊆ 𝐴
26		pwexr 7704	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ V)
27		elpwg 4552	. . . . . . 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 3930	. . . 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 2111  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4549  ∪ cuni 4858  dom cdm 5619   “ cima 5622  Ord word 6311  Oncon0 6312  suc csuc 6314  ‘cfv 6487  𝑅₁cr1 9661  rankcrnk 9662

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-r1 9663  df-rank 9664

This theorem is referenced by:  inatsk  10675