Theorem rankpw 3528

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		cleqid 1102	. . . . . . . 8 |- (rank` A) = (rank` A)
2		rankpw.1	. . . . . . . . . 10 |- A e. V
3	2	rankr1 3518	. . . . . . . . 9 |- ((rank` A) = (rank` A) <-> (-. A e. (R1` (rank` A)) /\ A e. (R1` suc (rank` A))))
4	3	pm3.26bd 259	. . . . . . . 8 |- ((rank` A) = (rank` A) -> -. A e. (R1` (rank` A)))
5	1, 4	ax-mp 6	. . . . . . 7 |- -. A e. (R1` (rank` A))
6	2	pwid 1805	. . . . . . . 8 |- A e. P~A
7		ssel 1502	. . . . . . . 8 |- (P~A (_ (R1` (rank` A)) -> (A e. P~A -> A e. (R1` (rank` A))))
8	6, 7	mpi 44	. . . . . . 7 |- (P~A (_ (R1` (rank` A)) -> A e. (R1` (rank` A)))
9	5, 8	mto 93	. . . . . 6 |- -. P~A (_ (R1` (rank` A))
10	2	pwex 1806	. . . . . . 7 |- P~A e. V
11	10	elpw 1801	. . . . . 6 |- (P~A e. P~(R1` (rank` A)) <-> P~A (_ (R1` (rank` A)))
12	9, 11	mtbir 167	. . . . 5 |- -. P~A e. P~(R1` (rank` A))
13		rankon 3515	. . . . . . 7 |- (rank` A) e. On
14		r1suc 3496	. . . . . . 7 |- ((rank` A) e. On -> (R1` suc (rank` A)) = P~(R1` (rank` A)))
15	13, 14	ax-mp 6	. . . . . 6 |- (R1` suc (rank` A)) = P~(R1` (rank` A))
16	15	eleq2i 1153	. . . . 5 |- (P~A e. (R1` suc (rank` A)) <-> P~A e. P~(R1` (rank` A)))
17	12, 16	mtbir 167	. . . 4 |- -. P~A e. (R1` suc (rank` A))
18	2	rankid 3516	. . . . . . . . . 10 |- A e. (R1` suc (rank` A))
19	18, 15	eleqtr 1161	. . . . . . . . 9 |- A e. P~(R1` (rank` A))
20	2	elpw 1801	. . . . . . . . 9 |- (A e. P~(R1` (rank` A)) <-> A (_ (R1` (rank` A)))
21	19, 20	mpbi 164	. . . . . . . 8 |- A (_ (R1` (rank` A))
22		sspwb 1863	. . . . . . . 8 |- (A (_ (R1` (rank` A)) <-> P~A (_ P~(R1` (rank` A)))
23	21, 22	mpbi 164	. . . . . . 7 |- P~A (_ P~(R1` (rank` A))
24	23, 15	sseqtr4 1533	. . . . . 6 |- P~A (_ (R1` suc (rank` A))
25	10	elpw 1801	. . . . . 6 |- (P~A e. P~(R1` suc (rank` A)) <-> P~A (_ (R1` suc (rank` A)))
26	24, 25	mpbir 165	. . . . 5 |- P~A e. P~(R1` suc (rank` A))
27	13	onsuc 2353	. . . . . 6 |- suc (rank` A) e. On
28		r1suc 3496	. . . . . 6 |- (suc (rank` A) e. On -> (R1` suc suc (rank` A)) = P~(R1` suc (rank` A)))
29	27, 28	ax-mp 6	. . . . 5 |- (R1` suc suc (rank` A)) = P~(R1` suc (rank` A))
30	26, 29	eleqtrr 1162	. . . 4 |- P~A e. (R1` suc suc (rank` A))
31	17, 30	pm3.2i 234	. . 3 |- (-. P~A e. (R1` suc (rank` A)) /\ P~A e. (R1` suc suc (rank` A)))
32	10	rankr1 3518	. . 3 |- (suc (rank` A) = (rank` P~A) <-> (-. P~A e. (R1` suc (rank` A)) /\ P~A e. (R1` suc suc (rank` A))))
33	31, 32	mpbir 165	. 2 |- suc (rank` A) = (rank` P~A)
34	33	cleqcomi 1105	1 |- (rank` P~A) = suc (rank` A)

Syntax hints:  -. wn 1   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  P~cpw 1798  Oncon0 2199  suc csuc 2201  ` cfv 2422  R1cr1 3485  rankcrnk 3486

This theorem is referenced by:  r1pw 3529  rankss 3531

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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