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Theorem rankpw 9837

Description: The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 22-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

**Hypothesis**
Ref	Expression
rankpw.1	⊢ 𝐴 ∈ V

**Assertion**
Ref	Expression
rankpw	⊢ (rank‘𝒫 𝐴) = suc (rank‘𝐴)

**Proof of Theorem rankpw**
Step	Hyp	Ref	Expression
1		rankpw.1	. . 3 ⊢ 𝐴 ∈ V
2		unir1 9807	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	eleqtrri 2832	. 2 ⊢ 𝐴 ∈ ∪ (𝑅₁ “ On)
4		rankpwi 9817	. 2 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
5	3, 4	ax-mp 5	1 ⊢ (rank‘𝒫 𝐴) = suc (rank‘𝐴)

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3474 𝒫 cpw 4602 ∪ cuni 4908 “ cima 5679 Oncon0 6364 suc csuc 6366 ‘cfv 6543 𝑅₁cr1 9756 rankcrnk 9757

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 7724 ax-reg 9586 ax-inf2 9635

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 7411 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-r1 9758 df-rank 9759

This theorem is referenced by: ranklim 9838 r1pwALT 9840 rankuni 9857 rankc2 9865 rankxpu 9870 rankmapu 9872 rankpwg 35136

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