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Theorem r1pw 8985
Description: A stronger property of 𝑅1 than rankpw 8983. 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 8963 . . . . . 6 (𝐴 (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
21eleq1d 2891 . . . . 5 (𝐴 (𝑅1 “ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
3 eloni 5973 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
4 ordsucelsuc 7283 . . . . . . 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 215 . . . . 5 (𝐵 ∈ On → (suc (rank‘𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
72, 6sylan9bb 507 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
8 pwwf 8947 . . . . . 6 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
98biimpi 208 . . . . 5 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
10 suceloni 7274 . . . . . 6 (𝐵 ∈ On → suc 𝐵 ∈ On)
11 r1fnon 8907 . . . . . . 7 𝑅1 Fn On
12 fndm 6223 . . . . . . 7 (𝑅1 Fn On → dom 𝑅1 = On)
1311, 12ax-mp 5 . . . . . 6 dom 𝑅1 = On
1410, 13syl6eleqr 2917 . . . . 5 (𝐵 ∈ On → suc 𝐵 ∈ dom 𝑅1)
15 rankr1ag 8942 . . . . 5 ((𝒫 𝐴 (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
169, 14, 15syl2an 591 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
1713eleq2i 2898 . . . . 5 (𝐵 ∈ dom 𝑅1𝐵 ∈ On)
18 rankr1ag 8942 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
1917, 18sylan2br 590 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
207, 16, 193bitr4rd 304 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
2120ex 403 . 2 (𝐴 (𝑅1 “ On) → (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
22 r1elwf 8936 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
23 r1elwf 8936 . . . . . 6 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 (𝑅1 “ On))
24 r1elssi 8945 . . . . . 6 (𝒫 𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
2523, 24syl 17 . . . . 5 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 (𝑅1 “ On))
26 ssid 3848 . . . . . 6 𝐴𝐴
27 pwexr 7234 . . . . . . 7 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ V)
28 elpwg 4386 . . . . . . 7 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐴𝐴𝐴))
2927, 28syl 17 . . . . . 6 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → (𝐴 ∈ 𝒫 𝐴𝐴𝐴))
3026, 29mpbiri 250 . . . . 5 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ 𝒫 𝐴)
3125, 30sseldd 3828 . . . 4 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 (𝑅1 “ On))
3222, 31pm5.21ni 369 . . 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 177 1 (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  Vcvv 3414  wss 3798  𝒫 cpw 4378   cuni 4658  dom cdm 5342  cima 5345  Ord word 5962  Oncon0 5963  suc csuc 5965   Fn wfn 6118  cfv 6123  𝑅1cr1 8902  rankcrnk 8903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-om 7327  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-r1 8904  df-rank 8905
This theorem is referenced by:  inatsk  9915
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