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Theorem r1pw 8652
Description: A stronger property of 𝑅1 than rankpw 8650. 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 8630 . . . . . 6 (𝐴 (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
21eleq1d 2683 . . . . 5 (𝐴 (𝑅1 “ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
3 eloni 5692 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
4 ordsucelsuc 6969 . . . . . . 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 213 . . . . 5 (𝐵 ∈ On → (suc (rank‘𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
72, 6sylan9bb 735 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
8 pwwf 8614 . . . . . 6 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
98biimpi 206 . . . . 5 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
10 suceloni 6960 . . . . . 6 (𝐵 ∈ On → suc 𝐵 ∈ On)
11 r1fnon 8574 . . . . . . 7 𝑅1 Fn On
12 fndm 5948 . . . . . . 7 (𝑅1 Fn On → dom 𝑅1 = On)
1311, 12ax-mp 5 . . . . . 6 dom 𝑅1 = On
1410, 13syl6eleqr 2709 . . . . 5 (𝐵 ∈ On → suc 𝐵 ∈ dom 𝑅1)
15 rankr1ag 8609 . . . . 5 ((𝒫 𝐴 (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
169, 14, 15syl2an 494 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
1713eleq2i 2690 . . . . 5 (𝐵 ∈ dom 𝑅1𝐵 ∈ On)
18 rankr1ag 8609 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
1917, 18sylan2br 493 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
207, 16, 193bitr4rd 301 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
2120ex 450 . 2 (𝐴 (𝑅1 “ On) → (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
22 r1elwf 8603 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
23 r1elwf 8603 . . . . . 6 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 (𝑅1 “ On))
24 r1elssi 8612 . . . . . 6 (𝒫 𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
2523, 24syl 17 . . . . 5 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 (𝑅1 “ On))
26 ssid 3603 . . . . . 6 𝐴𝐴
27 elex 3198 . . . . . . . 8 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 ∈ V)
28 pwexb 6922 . . . . . . . 8 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
2927, 28sylibr 224 . . . . . . 7 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ V)
30 elpwg 4138 . . . . . . 7 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐴𝐴𝐴))
3129, 30syl 17 . . . . . 6 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → (𝐴 ∈ 𝒫 𝐴𝐴𝐴))
3226, 31mpbiri 248 . . . . 5 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ 𝒫 𝐴)
3325, 32sseldd 3584 . . . 4 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 (𝑅1 “ On))
3422, 33pm5.21ni 367 . . 3 𝐴 (𝑅1 “ On) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
3534a1d 25 . 2 𝐴 (𝑅1 “ On) → (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
3621, 35pm2.61i 176 1 (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  wss 3555  𝒫 cpw 4130   cuni 4402  dom cdm 5074  cima 5077  Ord word 5681  Oncon0 5682  suc csuc 5684   Fn wfn 5842  cfv 5847  𝑅1cr1 8569  rankcrnk 8570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-r1 8571  df-rank 8572
This theorem is referenced by:  inatsk  9544
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