Theorem rankeq1o 32278

Description: The only set with rank 1o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)

Assertion

Ref	Expression
rankeq1o	|- ( ( rank ` A ) = 1o <-> A = { (/) } )

Proof of Theorem rankeq1o

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 7575	. . . . . . 7 |- 1o =/= (/)
2		neeq1 2856	. . . . . . 7 |- ( ( rank ` A ) = 1o -> ( ( rank ` A ) =/= (/) <-> 1o =/= (/) ) )
3	1, 2	mpbiri 248	. . . . . 6 |- ( ( rank ` A ) = 1o -> ( rank ` A ) =/= (/) )
4	3	neneqd 2799	. . . . 5 |- ( ( rank ` A ) = 1o -> -. ( rank ` A ) = (/) )
5		fvprc 6185	. . . . 5 |- ( -. A e. _V -> ( rank ` A ) = (/) )
6	4, 5	nsyl2 142	. . . 4 |- ( ( rank ` A ) = 1o -> A e. _V )
7		fveq2 6191	. . . . . . 7 |- ( x = A -> ( rank ` x ) = ( rank ` A ) )
8	7	eqeq1d 2624	. . . . . 6 |- ( x = A -> ( ( rank ` x ) = 1o <-> ( rank ` A ) = 1o ) )
9		eqeq1 2626	. . . . . 6 |- ( x = A -> ( x = 1o <-> A = 1o ) )
10	8, 9	imbi12d 334	. . . . 5 |- ( x = A -> ( ( ( rank ` x ) = 1o -> x = 1o ) <-> ( ( rank ` A ) = 1o -> A = 1o ) ) )
11		neeq1 2856	. . . . . . . 8 |- ( ( rank ` x ) = 1o -> ( ( rank ` x ) =/= (/) <-> 1o =/= (/) ) )
12	1, 11	mpbiri 248	. . . . . . 7 |- ( ( rank ` x ) = 1o -> ( rank ` x ) =/= (/) )
13		vex 3203	. . . . . . . . 9 |- x e. _V
14	13	rankeq0 8724	. . . . . . . 8 |- ( x = (/) <-> ( rank ` x ) = (/) )
15	14	necon3bii 2846	. . . . . . 7 |- ( x =/= (/) <-> ( rank ` x ) =/= (/) )
16	12, 15	sylibr 224	. . . . . 6 |- ( ( rank ` x ) = 1o -> x =/= (/) )
17	13	rankval 8679	. . . . . . . 8 |- ( rank ` x ) = |^| { y e. On | x e. ( R1 ` suc y ) }
18	17	eqeq1i 2627	. . . . . . 7 |- ( ( rank ` x ) = 1o <-> |^| { y e. On | x e. ( R1 ` suc y ) } = 1o )
19		ssrab2 3687	. . . . . . . . . . 11 |- { y e. On | x e. ( R1 ` suc y ) } C_ On
20		elirr 8505	. . . . . . . . . . . . . 14 |- -. 1o e. 1o
21		df1o2 7572	. . . . . . . . . . . . . . . 16 |- 1o = { (/) }
22		p0ex 4853	. . . . . . . . . . . . . . . 16 |- { (/) } e. _V
23	21, 22	eqeltri 2697	. . . . . . . . . . . . . . 15 |- 1o e. _V
24		id 22	. . . . . . . . . . . . . . 15 |- ( _V = 1o -> _V = 1o )
25	23, 24	syl5eleq 2707	. . . . . . . . . . . . . 14 |- ( _V = 1o -> 1o e. 1o )
26	20, 25	mto 188	. . . . . . . . . . . . 13 |- -. _V = 1o
27		inteq 4478	. . . . . . . . . . . . . . 15 |- ( { y e. On | x e. ( R1 ` suc y ) } = (/) -> |^| { y e. On | x e. ( R1 ` suc y ) } = |^| (/) )
28		int0 4490	. . . . . . . . . . . . . . 15 |- |^| (/) = _V
29	27, 28	syl6eq 2672	. . . . . . . . . . . . . 14 |- ( { y e. On | x e. ( R1 ` suc y ) } = (/) -> |^| { y e. On | x e. ( R1 ` suc y ) } = _V )
30	29	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( { y e. On | x e. ( R1 ` suc y ) } = (/) -> ( |^| { y e. On | x e. ( R1 ` suc y ) } = 1o <-> _V = 1o ) )
31	26, 30	mtbiri 317	. . . . . . . . . . . 12 |- ( { y e. On | x e. ( R1 ` suc y ) } = (/) -> -. |^| { y e. On | x e. ( R1 ` suc y ) } = 1o )
32	31	necon2ai 2823	. . . . . . . . . . 11 |- ( |^| { y e. On | x e. ( R1 ` suc y ) } = 1o -> { y e. On | x e. ( R1 ` suc y ) } =/= (/) )
33		onint 6995	. . . . . . . . . . 11 |- ( ( { y e. On | x e. ( R1 ` suc y ) } C_ On /\ { y e. On | x e. ( R1 ` suc y ) } =/= (/) ) -> |^| { y e. On | x e. ( R1 ` suc y ) } e. { y e. On | x e. ( R1 ` suc y ) } )
34	19, 32, 33	sylancr 695	. . . . . . . . . 10 |- ( |^| { y e. On | x e. ( R1 ` suc y ) } = 1o -> |^| { y e. On | x e. ( R1 ` suc y ) } e. { y e. On | x e. ( R1 ` suc y ) } )
35		eleq1 2689	. . . . . . . . . 10 |- ( |^| { y e. On | x e. ( R1 ` suc y ) } = 1o -> ( |^| { y e. On | x e. ( R1 ` suc y ) } e. { y e. On | x e. ( R1 ` suc y ) } <-> 1o e. { y e. On | x e. ( R1 ` suc y ) } ) )
36	34, 35	mpbid 222	. . . . . . . . 9 |- ( |^| { y e. On | x e. ( R1 ` suc y ) } = 1o -> 1o e. { y e. On | x e. ( R1 ` suc y ) } )
37		suceq 5790	. . . . . . . . . . . . 13 |- ( y = 1o -> suc y = suc 1o )
38	37	fveq2d 6195	. . . . . . . . . . . 12 |- ( y = 1o -> ( R1 ` suc y ) = ( R1 ` suc 1o ) )
39		df-1o 7560	. . . . . . . . . . . . . . . . 17 |- 1o = suc (/)
40	39	fveq2i 6194	. . . . . . . . . . . . . . . 16 |- ( R1 ` 1o ) = ( R1 ` suc (/) )
41		0elon 5778	. . . . . . . . . . . . . . . . 17 |- (/) e. On
42		r1suc 8633	. . . . . . . . . . . . . . . . 17 |- ( (/) e. On -> ( R1 ` suc (/) ) = ~P ( R1 ` (/) ) )
43	41, 42	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( R1 ` suc (/) ) = ~P ( R1 ` (/) )
44		r10 8631	. . . . . . . . . . . . . . . . 17 |- ( R1 ` (/) ) = (/)
45	44	pweqi 4162	. . . . . . . . . . . . . . . 16 |- ~P ( R1 ` (/) ) = ~P (/)
46	40, 43, 45	3eqtri 2648	. . . . . . . . . . . . . . 15 |- ( R1 ` 1o ) = ~P (/)
47	46	pweqi 4162	. . . . . . . . . . . . . 14 |- ~P ( R1 ` 1o ) = ~P ~P (/)
48		pw0 4343	. . . . . . . . . . . . . . 15 |- ~P (/) = { (/) }
49	48	pweqi 4162	. . . . . . . . . . . . . 14 |- ~P ~P (/) = ~P { (/) }
50		pwpw0 4344	. . . . . . . . . . . . . 14 |- ~P { (/) } = { (/) , { (/) } }
51	47, 49, 50	3eqtrri 2649	. . . . . . . . . . . . 13 |- { (/) , { (/) } } = ~P ( R1 ` 1o )
52		1on 7567	. . . . . . . . . . . . . 14 |- 1o e. On
53		r1suc 8633	. . . . . . . . . . . . . 14 |- ( 1o e. On -> ( R1 ` suc 1o ) = ~P ( R1 ` 1o ) )
54	52, 53	ax-mp 5	. . . . . . . . . . . . 13 |- ( R1 ` suc 1o ) = ~P ( R1 ` 1o )
55	51, 54	eqtr4i 2647	. . . . . . . . . . . 12 |- { (/) , { (/) } } = ( R1 ` suc 1o )
56	38, 55	syl6eqr 2674	. . . . . . . . . . 11 |- ( y = 1o -> ( R1 ` suc y ) = { (/) , { (/) } } )
57	56	eleq2d 2687	. . . . . . . . . 10 |- ( y = 1o -> ( x e. ( R1 ` suc y ) <-> x e. { (/) , { (/) } } ) )
58	57	elrab 3363	. . . . . . . . 9 |- ( 1o e. { y e. On | x e. ( R1 ` suc y ) } <-> ( 1o e. On /\ x e. { (/) , { (/) } } ) )
59	36, 58	sylib 208	. . . . . . . 8 |- ( |^| { y e. On | x e. ( R1 ` suc y ) } = 1o -> ( 1o e. On /\ x e. { (/) , { (/) } } ) )
60	13	elpr 4198	. . . . . . . . . 10 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
61		df-ne 2795	. . . . . . . . . . . 12 |- ( x =/= (/) <-> -. x = (/) )
62		orel1 397	. . . . . . . . . . . 12 |- ( -. x = (/) -> ( ( x = (/) \/ x = { (/) } ) -> x = { (/) } ) )
63	61, 62	sylbi 207	. . . . . . . . . . 11 |- ( x =/= (/) -> ( ( x = (/) \/ x = { (/) } ) -> x = { (/) } ) )
64		eqeq2 2633	. . . . . . . . . . . . 13 |- ( x = { (/) } -> ( 1o = x <-> 1o = { (/) } ) )
65	21, 64	mpbiri 248	. . . . . . . . . . . 12 |- ( x = { (/) } -> 1o = x )
66	65	eqcomd 2628	. . . . . . . . . . 11 |- ( x = { (/) } -> x = 1o )
67	63, 66	syl6com 37	. . . . . . . . . 10 |- ( ( x = (/) \/ x = { (/) } ) -> ( x =/= (/) -> x = 1o ) )
68	60, 67	sylbi 207	. . . . . . . . 9 |- ( x e. { (/) , { (/) } } -> ( x =/= (/) -> x = 1o ) )
69	68	adantl 482	. . . . . . . 8 |- ( ( 1o e. On /\ x e. { (/) , { (/) } } ) -> ( x =/= (/) -> x = 1o ) )
70	59, 69	syl 17	. . . . . . 7 |- ( |^| { y e. On | x e. ( R1 ` suc y ) } = 1o -> ( x =/= (/) -> x = 1o ) )
71	18, 70	sylbi 207	. . . . . 6 |- ( ( rank ` x ) = 1o -> ( x =/= (/) -> x = 1o ) )
72	16, 71	mpd 15	. . . . 5 |- ( ( rank ` x ) = 1o -> x = 1o )
73	10, 72	vtoclg 3266	. . . 4 |- ( A e. _V -> ( ( rank ` A ) = 1o -> A = 1o ) )
74	6, 73	mpcom 38	. . 3 |- ( ( rank ` A ) = 1o -> A = 1o )
75		fveq2 6191	. . . 4 |- ( A = 1o -> ( rank ` A ) = ( rank ` 1o ) )
76		r111 8638	. . . . . . 7 |- R1 : On -1-1-> _V
77		f1dm 6105	. . . . . . 7 |- ( R1 : On -1-1-> _V -> dom R1 = On )
78	76, 77	ax-mp 5	. . . . . 6 |- dom R1 = On
79	52, 78	eleqtrri 2700	. . . . 5 |- 1o e. dom R1
80		rankonid 8692	. . . . 5 |- ( 1o e. dom R1 <-> ( rank ` 1o ) = 1o )
81	79, 80	mpbi 220	. . . 4 |- ( rank ` 1o ) = 1o
82	75, 81	syl6eq 2672	. . 3 |- ( A = 1o -> ( rank ` A ) = 1o )
83	74, 82	impbii 199	. 2 |- ( ( rank ` A ) = 1o <-> A = 1o )
84	21	eqeq2i 2634	. 2 |- ( A = 1o <-> A = { (/) } )
85	83, 84	bitri 264	1 |- ( ( rank ` A ) = 1o <-> A = { (/) } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179   |^|cint 4475   dom cdm 5114   Oncon0 5723   suc csuc 5725   -1-1->wf1 5885   ` cfv 5888   1oc1o 7553   R1cr1 8625   rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-r1 8627  df-rank 8628

This theorem is referenced by: (None)