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Theorem rankpwi 9818

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 9817	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → ¬ 𝐴 ∈ (𝑅₁‘(rank‘𝐴)))
2		rankon 9790	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
3		r1suc 9765	. . . . . . 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 2826	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅₁‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ∈ 𝒫 (𝑅₁‘(rank‘𝐴)))
6		elpwi 4610	. . . . . 6 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅₁‘(rank‘𝐴)) → 𝒫 𝐴 ⊆ (𝑅₁‘(rank‘𝐴)))
7		pwidg 4623	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → 𝐴 ∈ 𝒫 𝐴)
8		ssel 3976	. . . . . 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 9799	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → 𝐴 ⊆ (𝑅₁‘(rank‘𝐴)))
13	12	sspwd 4616	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅₁‘(rank‘𝐴)))
14	13, 4	sseqtrrdi 4034	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → 𝒫 𝐴 ⊆ (𝑅₁‘suc (rank‘𝐴)))
15		fvex 6905	. . . . . 6 ⊢ (𝑅₁‘suc (rank‘𝐴)) ∈ V
16	15	elpw2 5346	. . . . 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 7827	. . . . 5 ⊢ suc (rank‘𝐴) ∈ On
19		r1suc 9765	. . . . 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 2845	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → 𝒫 𝐴 ∈ (𝑅₁‘suc suc (rank‘𝐴)))
22		pwwf 9802	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅₁ “ On))
23		rankr1c 9816	. . . 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 712	. 2 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → suc (rank‘𝐴) = (rank‘𝒫 𝐴))
26	25	eqcomd 2739	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 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3949 𝒫 cpw 4603 ∪ cuni 4909 “ cima 5680 Oncon0 6365 suc csuc 6367 ‘cfv 6544 𝑅₁cr1 9757 rankcrnk 9758

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-r1 9759 df-rank 9760

This theorem is referenced by: rankpw 9838 r1pw 9840 r1pwcl 9842

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