Theorem rankeq1o 36702

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 8478	. . . . . . 7 ⊢ 1o ≠ ∅
2		neeq1 3022	. . . . . . 7 ⊢ ((rank‘𝐴) = 1o → ((rank‘𝐴) ≠ ∅ ↔ 1o ≠ ∅))
3	1, 2	mpbiri 261	. . . . . 6 ⊢ ((rank‘𝐴) = 1o → (rank‘𝐴) ≠ ∅)
4	3	neneqd 2965	. . . . 5 ⊢ ((rank‘𝐴) = 1o → ¬ (rank‘𝐴) = ∅)
5		fvprc 6877	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅)
6	4, 5	nsyl2 142	. . . 4 ⊢ ((rank‘𝐴) = 1o → 𝐴 ∈ V)
7		fveqeq2 6894	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 1o ↔ (rank‘𝐴) = 1o))
8		eqeq1 2769	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 1o ↔ 𝐴 = 1o))
9	7, 8	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝐴 → (((rank‘𝑥) = 1o → 𝑥 = 1o) ↔ ((rank‘𝐴) = 1o → 𝐴 = 1o)))
10		neeq1 3022	. . . . . . . 8 ⊢ ((rank‘𝑥) = 1o → ((rank‘𝑥) ≠ ∅ ↔ 1o ≠ ∅))
11	1, 10	mpbiri 261	. . . . . . 7 ⊢ ((rank‘𝑥) = 1o → (rank‘𝑥) ≠ ∅)
12		vex 3461	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	rankeq0 9840	. . . . . . . 8 ⊢ (𝑥 = ∅ ↔ (rank‘𝑥) = ∅)
14	13	necon3bii 3012	. . . . . . 7 ⊢ (𝑥 ≠ ∅ ↔ (rank‘𝑥) ≠ ∅)
15	11, 14	sylibr 237	. . . . . 6 ⊢ ((rank‘𝑥) = 1o → 𝑥 ≠ ∅)
16	12	rankval 9795	. . . . . . . 8 ⊢ (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}
17	16	eqeq1i 2770	. . . . . . 7 ⊢ ((rank‘𝑥) = 1o ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
18		ssrab2 4035	. . . . . . . . . . 11 ⊢ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
19		elirr 9569	. . . . . . . . . . . . . 14 ⊢ ¬ 1o ∈ 1o
20		1oex 8469	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
21		id 23	. . . . . . . . . . . . . . 15 ⊢ (V = 1o → V = 1o)
22	20, 21	eleqtrid 2871	. . . . . . . . . . . . . 14 ⊢ (V = 1o → 1o ∈ 1o)
23	19, 22	mto 200	. . . . . . . . . . . . 13 ⊢ ¬ V = 1o
24		inteq 4917	. . . . . . . . . . . . . . 15 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∩ ∅)
25		int0 4929	. . . . . . . . . . . . . . 15 ⊢ ∩ ∅ = V
26	24, 25	eqtrdi 2816	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = V)
27	26	eqeq1d 2767	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o ↔ V = 1o))
28	23, 27	mtbiri 330	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ¬ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
29	28	necon2ai 2989	. . . . . . . . . . 11 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
30		onint 7795	. . . . . . . . . . 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)})
31	18, 29, 30	sylancr 599	. . . . . . . . . 10 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
32		eleq1 2853	. . . . . . . . . 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)}))
33	31, 32	mpbid 235	. . . . . . . . 9 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → 1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
34		suceq 6433	. . . . . . . . . . . . 13 ⊢ (𝑦 = 1o → suc 𝑦 = suc 1o)
35	34	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑦 = 1o → (𝑅1‘suc 𝑦) = (𝑅1‘suc 1o))
36		df-1o 8459	. . . . . . . . . . . . . . . . 17 ⊢ 1o = suc ∅
37	36	fveq2i 6888	. . . . . . . . . . . . . . . 16 ⊢ (𝑅1‘1o) = (𝑅1‘suc ∅)
38		0elon 6420	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ On
39		r1suc 9749	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅))
40	38, 39	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)
41		r10 9747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅1‘∅) = ∅
42	41	pweqi 4580	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 (𝑅1‘∅) = 𝒫 ∅
43	37, 40, 42	3eqtri 2792	. . . . . . . . . . . . . . 15 ⊢ (𝑅1‘1o) = 𝒫 ∅
44	43	pweqi 4580	. . . . . . . . . . . . . 14 ⊢ 𝒫 (𝑅1‘1o) = 𝒫 𝒫 ∅
45		pw0 4780	. . . . . . . . . . . . . . 15 ⊢ 𝒫 ∅ = {∅}
46	45	pweqi 4580	. . . . . . . . . . . . . 14 ⊢ 𝒫 𝒫 ∅ = 𝒫 {∅}
47		pwpw0 4781	. . . . . . . . . . . . . 14 ⊢ 𝒫 {∅} = {∅, {∅}}
48	44, 46, 47	3eqtrri 2793	. . . . . . . . . . . . 13 ⊢ {∅, {∅}} = 𝒫 (𝑅1‘1o)
49		1on 8472	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
50		r1suc 9749	. . . . . . . . . . . . . 14 ⊢ (1o ∈ On → (𝑅1‘suc 1o) = 𝒫 (𝑅1‘1o))
51	49, 50	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑅1‘suc 1o) = 𝒫 (𝑅1‘1o)
52	48, 51	eqtr4i 2791	. . . . . . . . . . . 12 ⊢ {∅, {∅}} = (𝑅1‘suc 1o)
53	35, 52	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝑦 = 1o → (𝑅1‘suc 𝑦) = {∅, {∅}})
54	53	eleq2d 2851	. . . . . . . . . 10 ⊢ (𝑦 = 1o → (𝑥 ∈ (𝑅1‘suc 𝑦) ↔ 𝑥 ∈ {∅, {∅}}))
55	54	elrab 3652	. . . . . . . . 9 ⊢ (1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
56	33, 55	sylib 221	. . . . . . . 8 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
57	12	elpr 4616	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
58		df-ne 2961	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
59		orel1 902	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
60	58, 59	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑥 ≠ ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
61		df1o2 8466	. . . . . . . . . . . . 13 ⊢ 1o = {∅}
62		eqeq2 2777	. . . . . . . . . . . . 13 ⊢ (𝑥 = {∅} → (1o = 𝑥 ↔ 1o = {∅}))
63	61, 62	mpbiri 261	. . . . . . . . . . . 12 ⊢ (𝑥 = {∅} → 1o = 𝑥)
64	63	eqcomd 2771	. . . . . . . . . . 11 ⊢ (𝑥 = {∅} → 𝑥 = 1o)
65	60, 64	syl6com 38	. . . . . . . . . 10 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
66	57, 65	sylbi 220	. . . . . . . . 9 ⊢ (𝑥 ∈ {∅, {∅}} → (𝑥 ≠ ∅ → 𝑥 = 1o))
67	66	adantl 487	. . . . . . . 8 ⊢ ((1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
68	56, 67	syl 18	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
69	17, 68	sylbi 220	. . . . . 6 ⊢ ((rank‘𝑥) = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
70	15, 69	mpd 16	. . . . 5 ⊢ ((rank‘𝑥) = 1o → 𝑥 = 1o)
71	9, 70	vtoclg 3524	. . . 4 ⊢ (𝐴 ∈ V → ((rank‘𝐴) = 1o → 𝐴 = 1o))
72	6, 71	mpcom 39	. . 3 ⊢ ((rank‘𝐴) = 1o → 𝐴 = 1o)
73		fveq2 6885	. . . 4 ⊢ (𝐴 = 1o → (rank‘𝐴) = (rank‘1o))
74		r111 9754	. . . . . . 7 ⊢ 𝑅1:On–1-1→V
75		f1dm 6784	. . . . . . 7 ⊢ (𝑅1:On–1-1→V → dom 𝑅1 = On)
76	74, 75	ax-mp 5	. . . . . 6 ⊢ dom 𝑅1 = On
77	49, 76	eleqtrri 2864	. . . . 5 ⊢ 1o ∈ dom 𝑅1
78		rankonid 9808	. . . . 5 ⊢ (1o ∈ dom 𝑅1 ↔ (rank‘1o) = 1o)
79	77, 78	mpbi 233	. . . 4 ⊢ (rank‘1o) = 1o
80	73, 79	eqtrdi 2816	. . 3 ⊢ (𝐴 = 1o → (rank‘𝐴) = 1o)
81	72, 80	impbii 212	. 2 ⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = 1o)
82	61	eqeq2i 2778	. 2 ⊢ (𝐴 = 1o ↔ 𝐴 = {∅})
83	81, 82	bitri 278	1 ⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = {∅})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  {crab 3418  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  {csn 4591  {cpr 4593  ∩ cint 4914  dom cdm 5663  Oncon0 6364  suc csuc 6366  –1-1→wf1 6537  ‘cfv 6540  1_oc1o 8452  𝑅₁cr1 9741  rankcrnk 9742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-reg 9561  ax-inf2 9617

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-r1 9743  df-rank 9744

This theorem is used by: (None)