Theorem rankpw 4664

Description: The rank of a power set. Part of Exercise 30 of [Enderton] p. 207.

Hypothesis

Ref	Expression
rankpw.1	|- A e. V

Assertion

Ref	Expression
rankpw	|- (rank` P~A) = suc (rank` A)

Proof of Theorem rankpw

Step	Hyp	Ref	Expression
1		rankpw.1	. . . . 5 |- A e. V
2	1	pwex 2740	. . . 4 |- P~A e. V
3	2	rankr1 4654	. . 3 |- (suc (rank` A) = (rank` P~A) <-> (-. P~A e. (R1` suc (rank` A)) /\ P~A e. (R1` suc suc (rank` A))))
4		eqid 1473	. . . . . . 7 |- (rank` A) = (rank` A)
5	1	rankr1 4654	. . . . . . . 8 |- ((rank` A) = (rank` A) <-> (-. A e. (R1` (rank` A)) /\ A e. (R1` suc (rank` A))))
6	5	pm3.26bi 322	. . . . . . 7 |- ((rank` A) = (rank` A) -> -. A e. (R1` (rank` A)))
7	4, 6	ax-mp 7	. . . . . 6 |- -. A e. (R1` (rank` A))
8	1	pwid 2404	. . . . . . 7 |- A e. P~A
9		ssel 2059	. . . . . . 7 |- (P~A (_ (R1` (rank` A)) -> (A e. P~A -> A e. (R1` (rank` A))))
10	8, 9	mpi 44	. . . . . 6 |- (P~A (_ (R1` (rank` A)) -> A e. (R1` (rank` A)))
11	7, 10	mto 106	. . . . 5 |- -. P~A (_ (R1` (rank` A))
12	2	elpw 2400	. . . . 5 |- (P~A e. P~(R1` (rank` A)) <-> P~A (_ (R1` (rank` A)))
13	11, 12	mtbir 192	. . . 4 |- -. P~A e. P~(R1` (rank` A))
14		rankon 4651	. . . . . 6 |- (rank` A) e. On
15		r1suc 4632	. . . . . 6 |- ((rank` A) e. On -> (R1` suc (rank` A)) = P~(R1` (rank` A)))
16	14, 15	ax-mp 7	. . . . 5 |- (R1` suc (rank` A)) = P~(R1` (rank` A))
17	16	eleq2i 1535	. . . 4 |- (P~A e. (R1` suc (rank` A)) <-> P~A e. P~(R1` (rank` A)))
18	13, 17	mtbir 192	. . 3 |- -. P~A e. (R1` suc (rank` A))
19	1	rankid 4652	. . . . . . . . 9 |- A e. (R1` suc (rank` A))
20	19, 16	eleqtr 1543	. . . . . . . 8 |- A e. P~(R1` (rank` A))
21	1	elpw 2400	. . . . . . . 8 |- (A e. P~(R1` (rank` A)) <-> A (_ (R1` (rank` A)))
22	20, 21	mpbi 189	. . . . . . 7 |- A (_ (R1` (rank` A))
23		sspwb 2750	. . . . . . 7 |- (A (_ (R1` (rank` A)) <-> P~A (_ P~(R1` (rank` A)))
24	22, 23	mpbi 189	. . . . . 6 |- P~A (_ P~(R1` (rank` A))
25	24, 16	sseqtr4 2090	. . . . 5 |- P~A (_ (R1` suc (rank` A))
26	2	elpw 2400	. . . . 5 |- (P~A e. P~(R1` suc (rank` A)) <-> P~A (_ (R1` suc (rank` A)))
27	25, 26	mpbir 190	. . . 4 |- P~A e. P~(R1` suc (rank` A))
28	14	onsuc 3100	. . . . 5 |- suc (rank` A) e. On
29		r1suc 4632	. . . . 5 |- (suc (rank` A) e. On -> (R1` suc suc (rank` A)) = P~(R1` suc (rank` A)))
30	28, 29	ax-mp 7	. . . 4 |- (R1` suc suc (rank` A)) = P~(R1` suc (rank` A))
31	27, 30	eleqtrr 1544	. . 3 |- P~A e. (R1` suc suc (rank` A))
32	3, 18, 31	mpbir2an 729	. 2 |- suc (rank` A) = (rank` P~A)
33	32	eqcomi 1476	1 |- (rank` P~A) = suc (rank` A)

Syntax hints:  -. wn 2   = wceq 954   e. wcel 956  Vcvv 1807   (_ wss 2043  P~cpw 2397  Oncon0 2943  suc csuc 2945  ` cfv 3177  R1cr1 4621  rankcrnk 4622

This theorem is referenced by:  ranklim 4665  r1pw 4666  rankss 4668  rankuni 4678  rankc2 4686  rankxpu 4691

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-r1 4623  df-rank 4624

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