MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankpwi Structured version   Visualization version   GIF version

Theorem rankpwi 9863
Description: The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 3-Jun-2013.)
Assertion
Ref Expression
rankpwi (𝐴 (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))

Proof of Theorem rankpwi
StepHypRef Expression
1 rankidn 9862 . . . 4 (𝐴 (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
2 rankon 9835 . . . . . . 7 (rank‘𝐴) ∈ On
3 r1suc 9810 . . . . . . 7 ((rank‘𝐴) ∈ On → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
42, 3ax-mp 5 . . . . . 6 (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))
54eleq2i 2833 . . . . 5 (𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
6 elpwi 4607 . . . . . 6 (𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) → 𝒫 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
7 pwidg 4620 . . . . . 6 (𝐴 (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
8 ssel 3977 . . . . . 6 (𝒫 𝐴 ⊆ (𝑅1‘(rank‘𝐴)) → (𝐴 ∈ 𝒫 𝐴𝐴 ∈ (𝑅1‘(rank‘𝐴))))
96, 7, 8syl2imc 41 . . . . 5 (𝐴 (𝑅1 “ On) → (𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
105, 9biimtrid 242 . . . 4 (𝐴 (𝑅1 “ On) → (𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
111, 10mtod 198 . . 3 (𝐴 (𝑅1 “ On) → ¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
12 r1rankidb 9844 . . . . . . 7 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
1312sspwd 4613 . . . . . 6 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
1413, 4sseqtrrdi 4025 . . . . 5 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
15 fvex 6919 . . . . . 6 (𝑅1‘suc (rank‘𝐴)) ∈ V
1615elpw2 5334 . . . . 5 (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
1714, 16sylibr 234 . . . 4 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
182onsuci 7859 . . . . 5 suc (rank‘𝐴) ∈ On
19 r1suc 9810 . . . . 5 (suc (rank‘𝐴) ∈ On → (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴)))
2018, 19ax-mp 5 . . . 4 (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴))
2117, 20eleqtrrdi 2852 . . 3 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
22 pwwf 9847 . . . 4 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
23 rankr1c 9861 . . . 4 (𝒫 𝐴 (𝑅1 “ On) → (suc (rank‘𝐴) = (rank‘𝒫 𝐴) ↔ (¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ∧ 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))))
2422, 23sylbi 217 . . 3 (𝐴 (𝑅1 “ On) → (suc (rank‘𝐴) = (rank‘𝒫 𝐴) ↔ (¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ∧ 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))))
2511, 21, 24mpbir2and 713 . 2 (𝐴 (𝑅1 “ On) → suc (rank‘𝐴) = (rank‘𝒫 𝐴))
2625eqcomd 2743 1 (𝐴 (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951  𝒫 cpw 4600   cuni 4907  cima 5688  Oncon0 6384  suc csuc 6386  cfv 6561  𝑅1cr1 9802  rankcrnk 9803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805
This theorem is referenced by:  rankpw  9883  r1pw  9885  r1pwcl  9887
  Copyright terms: Public domain W3C validator