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Theorem rankpwi 9820

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 9819	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → ¬ 𝐴 ∈ (𝑅₁‘(rank‘𝐴)))
2		rankon 9792	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
3		r1suc 9767	. . . . . . 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 2825	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅₁‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ∈ 𝒫 (𝑅₁‘(rank‘𝐴)))
6		elpwi 4609	. . . . . 6 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅₁‘(rank‘𝐴)) → 𝒫 𝐴 ⊆ (𝑅₁‘(rank‘𝐴)))
7		pwidg 4622	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → 𝐴 ∈ 𝒫 𝐴)
8		ssel 3975	. . . . . 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	biimtrid 241	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → (𝒫 𝐴 ∈ (𝑅₁‘suc (rank‘𝐴)) → 𝐴 ∈ (𝑅₁‘(rank‘𝐴))))
11	1, 10	mtod 197	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → ¬ 𝒫 𝐴 ∈ (𝑅₁‘suc (rank‘𝐴)))
12		r1rankidb 9801	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → 𝐴 ⊆ (𝑅₁‘(rank‘𝐴)))
13	12	sspwd 4615	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅₁‘(rank‘𝐴)))
14	13, 4	sseqtrrdi 4033	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → 𝒫 𝐴 ⊆ (𝑅₁‘suc (rank‘𝐴)))
15		fvex 6904	. . . . . 6 ⊢ (𝑅₁‘suc (rank‘𝐴)) ∈ V
16	15	elpw2 5345	. . . . 5 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅₁‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅₁‘suc (rank‘𝐴)))
17	14, 16	sylibr 233	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅₁‘suc (rank‘𝐴)))
18	2	onsuci 7829	. . . . 5 ⊢ suc (rank‘𝐴) ∈ On
19		r1suc 9767	. . . . 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 2844	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → 𝒫 𝐴 ∈ (𝑅₁‘suc suc (rank‘𝐴)))
22		pwwf 9804	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅₁ “ On))
23		rankr1c 9818	. . . 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 216	. . 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 2738	1 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3948 𝒫 cpw 4602 ∪ cuni 4908 “ cima 5679 Oncon0 6364 suc csuc 6366 ‘cfv 6543 𝑅₁cr1 9759 rankcrnk 9760

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-r1 9761 df-rank 9762

This theorem is referenced by: rankpw 9840 r1pw 9842 r1pwcl 9844

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