Theorem rankpwi 9251

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 9250	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
2		rankon 9223	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
3		r1suc 9198	. . . . . . 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 2904	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
6		elpwi 4547	. . . . . 6 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) → 𝒫 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
7		pwidg 4560	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
8		ssel 3960	. . . . . 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 244	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
11	1, 10	mtod 200	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
12		r1rankidb 9232	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
13	12	sspwd 4553	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
14	13, 4	sseqtrrdi 4017	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
15		fvex 6682	. . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) ∈ V
16	15	elpw2 5247	. . . . 5 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
17	14, 16	sylibr 236	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
18	2	onsuci 7552	. . . . 5 ⊢ suc (rank‘𝐴) ∈ On
19		r1suc 9198	. . . . 5 ⊢ (suc (rank‘𝐴) ∈ On → (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴)))
20	18, 19	ax-mp 5	. . . 4 ⊢ (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴))
21	17, 20	eleqtrrdi 2924	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
22		pwwf 9235	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
23		rankr1c 9249	. . . 4 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → (suc (rank‘𝐴) = (rank‘𝒫 𝐴) ↔ (¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ∧ 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))))
24	22, 23	sylbi 219	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (suc (rank‘𝐴) = (rank‘𝒫 𝐴) ↔ (¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ∧ 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))))
25	11, 21, 24	mpbir2and 711	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → suc (rank‘𝐴) = (rank‘𝒫 𝐴))
26	25	eqcomd 2827	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ⊆ wss 3935  𝒫 cpw 4538  ∪ cuni 4837   “ cima 5557  Oncon0 6190  suc csuc 6192  ‘cfv 6354  𝑅₁cr1 9190  rankcrnk 9191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-r1 9192  df-rank 9193

This theorem is referenced by:  rankpw  9271  r1pw  9273  r1pwcl  9275