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Theorem rankpwi 8630
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 8629 . . . 4 (𝐴 (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
2 rankon 8602 . . . . . . 7 (rank‘𝐴) ∈ On
3 r1suc 8577 . . . . . . 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 2690 . . . . 5 (𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
6 elpwi 4140 . . . . . 6 (𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) → 𝒫 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
7 pwidg 4144 . . . . . 6 (𝐴 (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
8 ssel 3577 . . . . . 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, 9syl5bi 232 . . . 4 (𝐴 (𝑅1 “ On) → (𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
111, 10mtod 189 . . 3 (𝐴 (𝑅1 “ On) → ¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
12 r1rankidb 8611 . . . . . . 7 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
13 sspwb 4878 . . . . . . 7 (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
1412, 13sylib 208 . . . . . 6 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
1514, 4syl6sseqr 3631 . . . . 5 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
16 fvex 6158 . . . . . 6 (𝑅1‘suc (rank‘𝐴)) ∈ V
1716elpw2 4788 . . . . 5 (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
1815, 17sylibr 224 . . . 4 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
192onsuci 6985 . . . . 5 suc (rank‘𝐴) ∈ On
20 r1suc 8577 . . . . 5 (suc (rank‘𝐴) ∈ On → (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴)))
2119, 20ax-mp 5 . . . 4 (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴))
2218, 21syl6eleqr 2709 . . 3 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
23 pwwf 8614 . . . 4 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
24 rankr1c 8628 . . . 4 (𝒫 𝐴 (𝑅1 “ On) → (suc (rank‘𝐴) = (rank‘𝒫 𝐴) ↔ (¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ∧ 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))))
2523, 24sylbi 207 . . 3 (𝐴 (𝑅1 “ On) → (suc (rank‘𝐴) = (rank‘𝒫 𝐴) ↔ (¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ∧ 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))))
2611, 22, 25mpbir2and 956 . 2 (𝐴 (𝑅1 “ On) → suc (rank‘𝐴) = (rank‘𝒫 𝐴))
2726eqcomd 2627 1 (𝐴 (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wss 3555  𝒫 cpw 4130   cuni 4402  cima 5077  Oncon0 5682  suc csuc 5684  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:  rankpw  8650  r1pw  8652  r1pwcl  8654
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