Theorem rankeq1o 32278

Description: The only set with rank 1_𝑜 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 ∈ V → (rank‘𝐴) = ∅)
6	4, 5	nsyl2 142	. . . 4 ⊢ ((rank‘𝐴) = 1𝑜 → 𝐴 ∈ 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 ∈ 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 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}
18	17	eqeq1i 2627	. . . . . . 7 ⊢ ((rank‘𝑥) = 1𝑜 ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜)
19		ssrab2 3687	. . . . . . . . . . 11 ⊢ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
20		elirr 8505	. . . . . . . . . . . . . 14 ⊢ ¬ 1𝑜 ∈ 1𝑜
21		df1o2 7572	. . . . . . . . . . . . . . . 16 ⊢ 1𝑜 = {∅}
22		p0ex 4853	. . . . . . . . . . . . . . . 16 ⊢ {∅} ∈ V
23	21, 22	eqeltri 2697	. . . . . . . . . . . . . . 15 ⊢ 1𝑜 ∈ V
24		id 22	. . . . . . . . . . . . . . 15 ⊢ (V = 1𝑜 → V = 1𝑜)
25	23, 24	syl5eleq 2707	. . . . . . . . . . . . . 14 ⊢ (V = 1𝑜 → 1𝑜 ∈ 1𝑜)
26	20, 25	mto 188	. . . . . . . . . . . . 13 ⊢ ¬ V = 1𝑜
27		inteq 4478	. . . . . . . . . . . . . . 15 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∩ ∅)
28		int0 4490	. . . . . . . . . . . . . . 15 ⊢ ∩ ∅ = V
29	27, 28	syl6eq 2672	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = V)
30	29	eqeq1d 2624	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 ↔ V = 1𝑜))
31	26, 30	mtbiri 317	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ¬ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜)
32	31	necon2ai 2823	. . . . . . . . . . 11 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
33		onint 6995	. . . . . . . . . . 11 ⊢ (({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
34	19, 32, 33	sylancr 695	. . . . . . . . . 10 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
35		eleq1 2689	. . . . . . . . . 10 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ 1𝑜 ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}))
36	34, 35	mpbid 222	. . . . . . . . 9 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → 1𝑜 ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (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 ⊢ ∅ ∈ On
42		r1suc 8633	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ 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 ∈ On
53		r1suc 8633	. . . . . . . . . . . . . 14 ⊢ (1𝑜 ∈ 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 ∈ (𝑅1‘suc 𝑦) ↔ 𝑥 ∈ {∅, {∅}}))
58	57	elrab 3363	. . . . . . . . 9 ⊢ (1𝑜 ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ (1𝑜 ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
59	36, 58	sylib 208	. . . . . . . 8 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → (1𝑜 ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
60	13	elpr 4198	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (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 ∈ {∅, {∅}} → (𝑥 ≠ ∅ → 𝑥 = 1𝑜))
69	68	adantl 482	. . . . . . . 8 ⊢ ((1𝑜 ∈ On ∧ 𝑥 ∈ {∅, {∅}}) → (𝑥 ≠ ∅ → 𝑥 = 1𝑜))
70	59, 69	syl 17	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (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 ∈ 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 ∈ dom 𝑅1
80		rankonid 8692	. . . . 5 ⊢ (1𝑜 ∈ 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   ∈ wcel 1990   ≠ wne 2794  {crab 2916  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  {cpr 4179  ∩ cint 4475  dom cdm 5114  Oncon0 5723  suc csuc 5725  –1-1→wf1 5885  ‘cfv 5888  1_𝑜c1o 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)