Theorem rankpwi 8686

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

Step	Hyp	Ref	Expression
1		rankidn 8685	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
2		rankon 8658	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
3		r1suc 8633	. . . . . . 7 ⊢ ((rank‘𝐴) ∈ On → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))
5	4	eleq2i 2693	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
6		elpwi 4168	. . . . . 6 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) → 𝒫 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
7		pwidg 4173	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
8		ssel 3597	. . . . . 6 ⊢ (𝒫 𝐴 ⊆ (𝑅1‘(rank‘𝐴)) → (𝐴 ∈ 𝒫 𝐴 → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
9	6, 7, 8	syl2imc 41	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
10	5, 9	syl5bi 232	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
11	1, 10	mtod 189	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
12		r1rankidb 8667	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
13		sspwb 4917	. . . . . . 7 ⊢ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
14	12, 13	sylib 208	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
15	14, 4	syl6sseqr 3652	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
16		fvex 6201	. . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) ∈ V
17	16	elpw2 4828	. . . . 5 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
18	15, 17	sylibr 224	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
19	2	onsuci 7038	. . . . 5 ⊢ suc (rank‘𝐴) ∈ On
20		r1suc 8633	. . . . 5 ⊢ (suc (rank‘𝐴) ∈ On → (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴)))
21	19, 20	ax-mp 5	. . . 4 ⊢ (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴))
22	18, 21	syl6eleqr 2712	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
23		pwwf 8670	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
24		rankr1c 8684	. . . 4 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → (suc (rank‘𝐴) = (rank‘𝒫 𝐴) ↔ (¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ∧ 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))))
25	23, 24	sylbi 207	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (suc (rank‘𝐴) = (rank‘𝒫 𝐴) ↔ (¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ∧ 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))))
26	11, 22, 25	mpbir2and 957	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → suc (rank‘𝐴) = (rank‘𝒫 𝐴))
27	26	eqcomd 2628	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436   “ cima 5117  Oncon0 5723  suc csuc 5725  ‘cfv 5888  𝑅₁cr1 8625  rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  rankpw  8706  r1pw  8708  r1pwcl  8710