Theorem rankeq1o 36932

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 8495	. . . . . . 7 ⊢ 1o ≠ ∅
2		neeq1 3018	. . . . . . 7 ⊢ ((rank‘𝐴) = 1o → ((rank‘𝐴) ≠ ∅ ↔ 1o ≠ ∅))
3	1, 2	mpbiri 261	. . . . . 6 ⊢ ((rank‘𝐴) = 1o → (rank‘𝐴) ≠ ∅)
4	3	neneqd 2961	. . . . 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 2765	. . . . . 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 3018	. . . . . . . 8 ⊢ ((rank‘𝑥) = 1o → ((rank‘𝑥) ≠ ∅ ↔ 1o ≠ ∅))
11	1, 10	mpbiri 261	. . . . . . 7 ⊢ ((rank‘𝑥) = 1o → (rank‘𝑥) ≠ ∅)
12		vex 3455	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	rankeq0 9877	. . . . . . . 8 ⊢ (𝑥 = ∅ ↔ (rank‘𝑥) = ∅)
14	13	necon3bii 3008	. . . . . . 7 ⊢ (𝑥 ≠ ∅ ↔ (rank‘𝑥) ≠ ∅)
15	11, 14	sylibr 237	. . . . . 6 ⊢ ((rank‘𝑥) = 1o → 𝑥 ≠ ∅)
16	12	rankval 9825	. . . . . . . 8 ⊢ (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}
17	16	eqeq1i 2766	. . . . . . 7 ⊢ ((rank‘𝑥) = 1o ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
18		ssrab2 4028	. . . . . . . . . . 11 ⊢ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
19		elirr 9594	. . . . . . . . . . . . . 14 ⊢ ¬ 1o ∈ 1o
20		1oex 8486	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
21		id 23	. . . . . . . . . . . . . . 15 ⊢ (V = 1o → V = 1o)
22	20, 21	eleqtrid 2867	. . . . . . . . . . . . . 14 ⊢ (V = 1o → 1o ∈ 1o)
23	19, 22	mto 200	. . . . . . . . . . . . 13 ⊢ ¬ V = 1o
24		inteq 4910	. . . . . . . . . . . . . . 15 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∩ ∅)
25		int0 4922	. . . . . . . . . . . . . . 15 ⊢ ∩ ∅ = V
26	24, 25	eqtrdi 2812	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = V)
27	26	eqeq1d 2763	. . . . . . . . . . . . 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 2985	. . . . . . . . . . 11 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
30		onint 7804	. . . . . . . . . . 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 2849	. . . . . . . . . 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 6431	. . . . . . . . . . . . 13 ⊢ (𝑦 = 1o → suc 𝑦 = suc 1o)
35	34	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑦 = 1o → (𝑅1‘suc 𝑦) = (𝑅1‘suc 1o))
36		df-1o 8476	. . . . . . . . . . . . . . . . 17 ⊢ 1o = suc ∅
37	36	fveq2i 6888	. . . . . . . . . . . . . . . 16 ⊢ (𝑅1‘1o) = (𝑅1‘suc ∅)
38		0elon 6418	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ On
39		r1suc 9777	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅))
40	38, 39	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)
41		r10 9775	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅1‘∅) = ∅
42	41	pweqi 4573	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 (𝑅1‘∅) = 𝒫 ∅
43	37, 40, 42	3eqtri 2788	. . . . . . . . . . . . . . 15 ⊢ (𝑅1‘1o) = 𝒫 ∅
44	43	pweqi 4573	. . . . . . . . . . . . . 14 ⊢ 𝒫 (𝑅1‘1o) = 𝒫 𝒫 ∅
45		pw0 4773	. . . . . . . . . . . . . . 15 ⊢ 𝒫 ∅ = {∅}
46	45	pweqi 4573	. . . . . . . . . . . . . 14 ⊢ 𝒫 𝒫 ∅ = 𝒫 {∅}
47		pwpw0 4774	. . . . . . . . . . . . . 14 ⊢ 𝒫 {∅} = {∅, {∅}}
48	44, 46, 47	3eqtrri 2789	. . . . . . . . . . . . 13 ⊢ {∅, {∅}} = 𝒫 (𝑅1‘1o)
49		1on 8489	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
50		r1suc 9777	. . . . . . . . . . . . . 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 2787	. . . . . . . . . . . 12 ⊢ {∅, {∅}} = (𝑅1‘suc 1o)
53	35, 52	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝑦 = 1o → (𝑅1‘suc 𝑦) = {∅, {∅}})
54	53	eleq2d 2847	. . . . . . . . . 10 ⊢ (𝑦 = 1o → (𝑥 ∈ (𝑅1‘suc 𝑦) ↔ 𝑥 ∈ {∅, {∅}}))
55	54	elrab 3645	. . . . . . . . 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 4609	. . . . . . . . 9 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
58		df-ne 2957	. . . . . . . . . . 11 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
59		orel1 902	. . . . . . . . . . 11 ⊢ (¬ 𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
60	58, 59	sylbi 220	. . . . . . . . . 10 ⊢ (𝑥 ≠ ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
61		df1o2 8483	. . . . . . . . . . . 12 ⊢ 1o = {∅}
62		eqeq2 2773	. . . . . . . . . . . 12 ⊢ (𝑥 = {∅} → (1o = 𝑥 ↔ 1o = {∅}))
63	61, 62	mpbiri 261	. . . . . . . . . . 11 ⊢ (𝑥 = {∅} → 1o = 𝑥)
64	63	eqcomd 2767	. . . . . . . . . 10 ⊢ (𝑥 = {∅} → 𝑥 = 1o)
65	60, 64	syl6com 38	. . . . . . . . 9 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
66	57, 65	sylbi 220	. . . . . . . 8 ⊢ (𝑥 ∈ {∅, {∅}} → (𝑥 ≠ ∅ → 𝑥 = 1o))
67	56, 66	simpl2im 513	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
68	17, 67	sylbi 220	. . . . . 6 ⊢ ((rank‘𝑥) = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
69	15, 68	mpd 16	. . . . 5 ⊢ ((rank‘𝑥) = 1o → 𝑥 = 1o)
70	9, 69	vtoclg 3518	. . . 4 ⊢ (𝐴 ∈ V → ((rank‘𝐴) = 1o → 𝐴 = 1o))
71	6, 70	mpcom 39	. . 3 ⊢ ((rank‘𝐴) = 1o → 𝐴 = 1o)
72		fveq2 6885	. . . 4 ⊢ (𝐴 = 1o → (rank‘𝐴) = (rank‘1o))
73		r1fnon 9773	. . . . . . 7 ⊢ 𝑅1 Fn On
74	73	fndmi 6643	. . . . . 6 ⊢ dom 𝑅1 = On
75	49, 74	eleqtrri 2860	. . . . 5 ⊢ 1o ∈ dom 𝑅1
76		rankonid 9839	. . . . 5 ⊢ (1o ∈ dom 𝑅1 ↔ (rank‘1o) = 1o)
77	75, 76	mpbi 233	. . . 4 ⊢ (rank‘1o) = 1o
78	72, 77	eqtrdi 2812	. . 3 ⊢ (𝐴 = 1o → (rank‘𝐴) = 1o)
79	71, 78	impbii 212	. 2 ⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = 1o)
80	61	eqeq2i 2774	. 2 ⊢ (𝐴 = 1o ↔ 𝐴 = {∅})
81	79, 80	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 2145   ≠ wne 2956  {crab 3413  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  {cpr 4586  ∩ cint 4907  dom cdm 5651  Oncon0 6362  suc csuc 6364  ‘cfv 6538  1_oc1o 8469  𝑅₁cr1 9766  rankcrnk 9767

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-reg 9586  ax-inf2 9642

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-r1 9768  df-rank 9769

This theorem is used by: (None)