Theorem rankeq1o 34400

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 8286	. . . . . . 7 ⊢ 1o ≠ ∅
2		neeq1 3005	. . . . . . 7 ⊢ ((rank‘𝐴) = 1o → ((rank‘𝐴) ≠ ∅ ↔ 1o ≠ ∅))
3	1, 2	mpbiri 257	. . . . . 6 ⊢ ((rank‘𝐴) = 1o → (rank‘𝐴) ≠ ∅)
4	3	neneqd 2947	. . . . 5 ⊢ ((rank‘𝐴) = 1o → ¬ (rank‘𝐴) = ∅)
5		fvprc 6748	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅)
6	4, 5	nsyl2 141	. . . 4 ⊢ ((rank‘𝐴) = 1o → 𝐴 ∈ V)
7		fveqeq2 6765	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 1o ↔ (rank‘𝐴) = 1o))
8		eqeq1 2742	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 1o ↔ 𝐴 = 1o))
9	7, 8	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝐴 → (((rank‘𝑥) = 1o → 𝑥 = 1o) ↔ ((rank‘𝐴) = 1o → 𝐴 = 1o)))
10		neeq1 3005	. . . . . . . 8 ⊢ ((rank‘𝑥) = 1o → ((rank‘𝑥) ≠ ∅ ↔ 1o ≠ ∅))
11	1, 10	mpbiri 257	. . . . . . 7 ⊢ ((rank‘𝑥) = 1o → (rank‘𝑥) ≠ ∅)
12		vex 3426	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	rankeq0 9550	. . . . . . . 8 ⊢ (𝑥 = ∅ ↔ (rank‘𝑥) = ∅)
14	13	necon3bii 2995	. . . . . . 7 ⊢ (𝑥 ≠ ∅ ↔ (rank‘𝑥) ≠ ∅)
15	11, 14	sylibr 233	. . . . . 6 ⊢ ((rank‘𝑥) = 1o → 𝑥 ≠ ∅)
16	12	rankval 9505	. . . . . . . 8 ⊢ (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}
17	16	eqeq1i 2743	. . . . . . 7 ⊢ ((rank‘𝑥) = 1o ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
18		ssrab2 4009	. . . . . . . . . . 11 ⊢ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
19		elirr 9286	. . . . . . . . . . . . . 14 ⊢ ¬ 1o ∈ 1o
20		1oex 8280	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
21		id 22	. . . . . . . . . . . . . . 15 ⊢ (V = 1o → V = 1o)
22	20, 21	eleqtrid 2845	. . . . . . . . . . . . . 14 ⊢ (V = 1o → 1o ∈ 1o)
23	19, 22	mto 196	. . . . . . . . . . . . 13 ⊢ ¬ V = 1o
24		inteq 4879	. . . . . . . . . . . . . . 15 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∩ ∅)
25		int0 4890	. . . . . . . . . . . . . . 15 ⊢ ∩ ∅ = V
26	24, 25	eqtrdi 2795	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = V)
27	26	eqeq1d 2740	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o ↔ V = 1o))
28	23, 27	mtbiri 326	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ¬ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
29	28	necon2ai 2972	. . . . . . . . . . 11 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
30		onint 7617	. . . . . . . . . . 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 586	. . . . . . . . . 10 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
32		eleq1 2826	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → 1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
34		suceq 6316	. . . . . . . . . . . . 13 ⊢ (𝑦 = 1o → suc 𝑦 = suc 1o)
35	34	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (𝑦 = 1o → (𝑅1‘suc 𝑦) = (𝑅1‘suc 1o))
36		df-1o 8267	. . . . . . . . . . . . . . . . 17 ⊢ 1o = suc ∅
37	36	fveq2i 6759	. . . . . . . . . . . . . . . 16 ⊢ (𝑅1‘1o) = (𝑅1‘suc ∅)
38		0elon 6304	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ On
39		r1suc 9459	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅))
40	38, 39	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)
41		r10 9457	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅1‘∅) = ∅
42	41	pweqi 4548	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 (𝑅1‘∅) = 𝒫 ∅
43	37, 40, 42	3eqtri 2770	. . . . . . . . . . . . . . 15 ⊢ (𝑅1‘1o) = 𝒫 ∅
44	43	pweqi 4548	. . . . . . . . . . . . . 14 ⊢ 𝒫 (𝑅1‘1o) = 𝒫 𝒫 ∅
45		pw0 4742	. . . . . . . . . . . . . . 15 ⊢ 𝒫 ∅ = {∅}
46	45	pweqi 4548	. . . . . . . . . . . . . 14 ⊢ 𝒫 𝒫 ∅ = 𝒫 {∅}
47		pwpw0 4743	. . . . . . . . . . . . . 14 ⊢ 𝒫 {∅} = {∅, {∅}}
48	44, 46, 47	3eqtrri 2771	. . . . . . . . . . . . 13 ⊢ {∅, {∅}} = 𝒫 (𝑅1‘1o)
49		1on 8274	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
50		r1suc 9459	. . . . . . . . . . . . . 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 2769	. . . . . . . . . . . 12 ⊢ {∅, {∅}} = (𝑅1‘suc 1o)
53	35, 52	eqtr4di 2797	. . . . . . . . . . 11 ⊢ (𝑦 = 1o → (𝑅1‘suc 𝑦) = {∅, {∅}})
54	53	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝑦 = 1o → (𝑥 ∈ (𝑅1‘suc 𝑦) ↔ 𝑥 ∈ {∅, {∅}}))
55	54	elrab 3617	. . . . . . . . 9 ⊢ (1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
56	33, 55	sylib 217	. . . . . . . 8 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
57	12	elpr 4581	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
58		df-ne 2943	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
59		orel1 885	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
60	58, 59	sylbi 216	. . . . . . . . . . 11 ⊢ (𝑥 ≠ ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
61		df1o2 8279	. . . . . . . . . . . . 13 ⊢ 1o = {∅}
62		eqeq2 2750	. . . . . . . . . . . . 13 ⊢ (𝑥 = {∅} → (1o = 𝑥 ↔ 1o = {∅}))
63	61, 62	mpbiri 257	. . . . . . . . . . . 12 ⊢ (𝑥 = {∅} → 1o = 𝑥)
64	63	eqcomd 2744	. . . . . . . . . . 11 ⊢ (𝑥 = {∅} → 𝑥 = 1o)
65	60, 64	syl6com 37	. . . . . . . . . 10 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
66	57, 65	sylbi 216	. . . . . . . . 9 ⊢ (𝑥 ∈ {∅, {∅}} → (𝑥 ≠ ∅ → 𝑥 = 1o))
67	66	adantl 481	. . . . . . . 8 ⊢ ((1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
68	56, 67	syl 17	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
69	17, 68	sylbi 216	. . . . . 6 ⊢ ((rank‘𝑥) = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
70	15, 69	mpd 15	. . . . 5 ⊢ ((rank‘𝑥) = 1o → 𝑥 = 1o)
71	9, 70	vtoclg 3495	. . . 4 ⊢ (𝐴 ∈ V → ((rank‘𝐴) = 1o → 𝐴 = 1o))
72	6, 71	mpcom 38	. . 3 ⊢ ((rank‘𝐴) = 1o → 𝐴 = 1o)
73		fveq2 6756	. . . 4 ⊢ (𝐴 = 1o → (rank‘𝐴) = (rank‘1o))
74		r111 9464	. . . . . . 7 ⊢ 𝑅1:On–1-1→V
75		f1dm 6658	. . . . . . 7 ⊢ (𝑅1:On–1-1→V → dom 𝑅1 = On)
76	74, 75	ax-mp 5	. . . . . 6 ⊢ dom 𝑅1 = On
77	49, 76	eleqtrri 2838	. . . . 5 ⊢ 1o ∈ dom 𝑅1
78		rankonid 9518	. . . . 5 ⊢ (1o ∈ dom 𝑅1 ↔ (rank‘1o) = 1o)
79	77, 78	mpbi 229	. . . 4 ⊢ (rank‘1o) = 1o
80	73, 79	eqtrdi 2795	. . 3 ⊢ (𝐴 = 1o → (rank‘𝐴) = 1o)
81	72, 80	impbii 208	. 2 ⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = 1o)
82	61	eqeq2i 2751	. 2 ⊢ (𝐴 = 1o ↔ 𝐴 = {∅})
83	81, 82	bitri 274	1 ⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = {∅})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  {crab 3067  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558  {cpr 4560  ∩ cint 4876  dom cdm 5580  Oncon0 6251  suc csuc 6253  –1-1→wf1 6415  ‘cfv 6418  1_oc1o 8260  𝑅₁cr1 9451  rankcrnk 9452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-r1 9453  df-rank 9454

This theorem is referenced by: (None)