Theorem rankpwi 9856

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 9855	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
2		rankon 9827	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
3		r1suc 9801	. . . . . . 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 2856	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
6		elpwi 4567	. . . . . 6 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) → 𝒫 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
7		pwidg 4581	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
8		ssel 3907	. . . . . 6 ⊢ (𝒫 𝐴 ⊆ (𝑅1‘(rank‘𝐴)) → (𝐴 ∈ 𝒫 𝐴 → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
9	6, 7, 8	syl2imc 42	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝒫 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
10	5, 9	biimtrid 245	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
11	1, 10	mtod 201	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝒫 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
12		r1rankidb 9836	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
13	12	sspwd 4573	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
14	13, 4	sseqtrrdi 3954	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
15		fvex 6918	. . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) ∈ V
16	15	elpw2 5300	. . . . 5 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
17	14, 16	sylibr 237	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
18	2	onsuci 7871	. . . . 5 ⊢ suc (rank‘𝐴) ∈ On
19		r1suc 9801	. . . . 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 2875	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
22		pwwf 9839	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
23		rankr1c 9854	. . . 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 220	. . 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 726	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → suc (rank‘𝐴) = (rank‘𝒫 𝐴))
26	25	eqcomd 2769	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  𝒫 cpw 4560  ∪ cuni 4870   “ cima 5658  Oncon0 6377  suc csuc 6379  ‘cfv 6555  𝑅₁cr1 9790  rankcrnk 9791

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-r1 9792  df-rank 9793

This theorem is used by:  rankpw  9880  r1pw  9883  r1pwcl  9885