Theorem r1pw 9757

Description: A stronger property of 𝑅₁ than rankpw 9755. 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 9735	. . . . . 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 6327	. . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordsucelsuc 7764	. . . . . . 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 9719	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
9	8	biimpi 216	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
10		onsuc 7755	. . . . . 6 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
11		r1fnon 9679	. . . . . . 7 ⊢ 𝑅1 Fn On
12	11	fndmi 6596	. . . . . 6 ⊢ dom 𝑅1 = On
13	10, 12	eleqtrrdi 2847	. . . . 5 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ dom 𝑅1)
14		rankr1ag 9714	. . . . 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 2828	. . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On)
17		rankr1ag 9714	. . . . 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 9708	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
22		r1elwf 9708	. . . . . 6 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
23		r1elssi 9717	. . . . . 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 3956	. . . . . 6 ⊢ 𝐴 ⊆ 𝐴
26		pwexr 7710	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ V)
27		elpwg 4557	. . . . . . 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 3934	. . . 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 2113  Vcvv 3440   ⊆ wss 3901  𝒫 cpw 4554  ∪ cuni 4863  dom cdm 5624   “ cima 5627  Ord word 6316  Oncon0 6317  suc csuc 6319  ‘cfv 6492  𝑅₁cr1 9674  rankcrnk 9675

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-r1 9676  df-rank 9677

This theorem is referenced by:  inatsk  10689