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Theorem rankpw 9858

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 9828	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	eleqtrri 2827	. 2 ⊢ 𝐴 ∈ ∪ (𝑅₁ “ On)
4		rankpwi 9838	. 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 1534 ∈ wcel 2099 Vcvv 3469 𝒫 cpw 4598 ∪ cuni 4903 “ cima 5675 Oncon0 6363 suc csuc 6365 ‘cfv 6542 𝑅₁cr1 9777 rankcrnk 9778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-reg 9607 ax-inf2 9656

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-r1 9779 df-rank 9780

This theorem is referenced by: ranklim 9859 r1pwALT 9861 rankuni 9878 rankc2 9886 rankxpu 9891 rankmapu 9893 rankpwg 35701

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