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Theorem r1pw 9260
Description: A stronger property of 𝑅1 than rankpw 9258. 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 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))

Proof of Theorem r1pw
StepHypRef Expression
1 rankpwi 9238 . . . . . 6 (𝐴 (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
21eleq1d 2900 . . . . 5 (𝐴 (𝑅1 “ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
3 eloni 6184 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
4 ordsucelsuc 7522 . . . . . . 7 (Ord 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
53, 4syl 17 . . . . . 6 (𝐵 ∈ On → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
65bicomd 226 . . . . 5 (𝐵 ∈ On → (suc (rank‘𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
72, 6sylan9bb 513 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
8 pwwf 9222 . . . . . 6 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
98biimpi 219 . . . . 5 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
10 suceloni 7513 . . . . . 6 (𝐵 ∈ On → suc 𝐵 ∈ On)
11 r1fnon 9182 . . . . . . 7 𝑅1 Fn On
12 fndm 6438 . . . . . . 7 (𝑅1 Fn On → dom 𝑅1 = On)
1311, 12ax-mp 5 . . . . . 6 dom 𝑅1 = On
1410, 13eleqtrrdi 2927 . . . . 5 (𝐵 ∈ On → suc 𝐵 ∈ dom 𝑅1)
15 rankr1ag 9217 . . . . 5 ((𝒫 𝐴 (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
169, 14, 15syl2an 598 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
1713eleq2i 2907 . . . . 5 (𝐵 ∈ dom 𝑅1𝐵 ∈ On)
18 rankr1ag 9217 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
1917, 18sylan2br 597 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
207, 16, 193bitr4rd 315 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
2120ex 416 . 2 (𝐴 (𝑅1 “ On) → (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
22 r1elwf 9211 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
23 r1elwf 9211 . . . . . 6 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 (𝑅1 “ On))
24 r1elssi 9220 . . . . . 6 (𝒫 𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
2523, 24syl 17 . . . . 5 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 (𝑅1 “ On))
26 ssid 3973 . . . . . 6 𝐴𝐴
27 pwexr 7472 . . . . . . 7 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ V)
28 elpwg 4523 . . . . . . 7 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐴𝐴𝐴))
2927, 28syl 17 . . . . . 6 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → (𝐴 ∈ 𝒫 𝐴𝐴𝐴))
3026, 29mpbiri 261 . . . . 5 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ 𝒫 𝐴)
3125, 30sseldd 3952 . . . 4 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 (𝑅1 “ On))
3222, 31pm5.21ni 382 . . 3 𝐴 (𝑅1 “ On) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
3332a1d 25 . 2 𝐴 (𝑅1 “ On) → (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
3421, 33pm2.61i 185 1 (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  Vcvv 3479  wss 3918  𝒫 cpw 4520   cuni 4821  dom cdm 5538  cima 5541  Ord word 6173  Oncon0 6174  suc csuc 6176   Fn wfn 6333  cfv 6338  𝑅1cr1 9177  rankcrnk 9178
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-int 4860  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-om 7566  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-r1 9179  df-rank 9180
This theorem is referenced by:  inatsk  10187
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