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Theorem r1pw 8958
Description: A stronger property of 𝑅1 than rankpw 8956. 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 8936 . . . . . 6 (𝐴 (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
21eleq1d 2863 . . . . 5 (𝐴 (𝑅1 “ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
3 eloni 5951 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
4 ordsucelsuc 7256 . . . . . . 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 506 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
8 pwwf 8920 . . . . . 6 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
98biimpi 208 . . . . 5 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
10 suceloni 7247 . . . . . 6 (𝐵 ∈ On → suc 𝐵 ∈ On)
11 r1fnon 8880 . . . . . . 7 𝑅1 Fn On
12 fndm 6201 . . . . . . 7 (𝑅1 Fn On → dom 𝑅1 = On)
1311, 12ax-mp 5 . . . . . 6 dom 𝑅1 = On
1410, 13syl6eleqr 2889 . . . . 5 (𝐵 ∈ On → suc 𝐵 ∈ dom 𝑅1)
15 rankr1ag 8915 . . . . 5 ((𝒫 𝐴 (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
169, 14, 15syl2an 590 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
1713eleq2i 2870 . . . . 5 (𝐵 ∈ dom 𝑅1𝐵 ∈ On)
18 rankr1ag 8915 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
1917, 18sylan2br 589 . . . 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 402 . 2 (𝐴 (𝑅1 “ On) → (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
22 r1elwf 8909 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
23 r1elwf 8909 . . . . . 6 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 (𝑅1 “ On))
24 r1elssi 8918 . . . . . 6 (𝒫 𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
2523, 24syl 17 . . . . 5 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 (𝑅1 “ On))
26 ssid 3819 . . . . . 6 𝐴𝐴
27 pwexr 7207 . . . . . . 7 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ V)
28 elpwg 4357 . . . . . . 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 3799 . . . 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 385   = wceq 1653  wcel 2157  Vcvv 3385  wss 3769  𝒫 cpw 4349   cuni 4628  dom cdm 5312  cima 5315  Ord word 5940  Oncon0 5941  suc csuc 5943   Fn wfn 6096  cfv 6101  𝑅1cr1 8875  rankcrnk 8876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-r1 8877  df-rank 8878
This theorem is referenced by:  inatsk  9888
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