Theorem rankeq1o 36365

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 8415	. . . . . . 7 ⊢ 1o ≠ ∅
2		neeq1 2994	. . . . . . 7 ⊢ ((rank‘𝐴) = 1o → ((rank‘𝐴) ≠ ∅ ↔ 1o ≠ ∅))
3	1, 2	mpbiri 258	. . . . . 6 ⊢ ((rank‘𝐴) = 1o → (rank‘𝐴) ≠ ∅)
4	3	neneqd 2937	. . . . 5 ⊢ ((rank‘𝐴) = 1o → ¬ (rank‘𝐴) = ∅)
5		fvprc 6826	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅)
6	4, 5	nsyl2 141	. . . 4 ⊢ ((rank‘𝐴) = 1o → 𝐴 ∈ V)
7		fveqeq2 6843	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 1o ↔ (rank‘𝐴) = 1o))
8		eqeq1 2740	. . . . . 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 2994	. . . . . . . 8 ⊢ ((rank‘𝑥) = 1o → ((rank‘𝑥) ≠ ∅ ↔ 1o ≠ ∅))
11	1, 10	mpbiri 258	. . . . . . 7 ⊢ ((rank‘𝑥) = 1o → (rank‘𝑥) ≠ ∅)
12		vex 3444	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13	12	rankeq0 9773	. . . . . . . 8 ⊢ (𝑥 = ∅ ↔ (rank‘𝑥) = ∅)
14	13	necon3bii 2984	. . . . . . 7 ⊢ (𝑥 ≠ ∅ ↔ (rank‘𝑥) ≠ ∅)
15	11, 14	sylibr 234	. . . . . 6 ⊢ ((rank‘𝑥) = 1o → 𝑥 ≠ ∅)
16	12	rankval 9728	. . . . . . . 8 ⊢ (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}
17	16	eqeq1i 2741	. . . . . . 7 ⊢ ((rank‘𝑥) = 1o ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
18		ssrab2 4032	. . . . . . . . . . 11 ⊢ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
19		elirr 9504	. . . . . . . . . . . . . 14 ⊢ ¬ 1o ∈ 1o
20		1oex 8407	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
21		id 22	. . . . . . . . . . . . . . 15 ⊢ (V = 1o → V = 1o)
22	20, 21	eleqtrid 2842	. . . . . . . . . . . . . 14 ⊢ (V = 1o → 1o ∈ 1o)
23	19, 22	mto 197	. . . . . . . . . . . . 13 ⊢ ¬ V = 1o
24		inteq 4905	. . . . . . . . . . . . . . 15 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∩ ∅)
25		int0 4917	. . . . . . . . . . . . . . 15 ⊢ ∩ ∅ = V
26	24, 25	eqtrdi 2787	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = V)
27	26	eqeq1d 2738	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o ↔ V = 1o))
28	23, 27	mtbiri 327	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ¬ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
29	28	necon2ai 2961	. . . . . . . . . . 11 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
30		onint 7735	. . . . . . . . . . 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 587	. . . . . . . . . 10 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
32		eleq1 2824	. . . . . . . . . 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 232	. . . . . . . . 9 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → 1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
34		suceq 6385	. . . . . . . . . . . . 13 ⊢ (𝑦 = 1o → suc 𝑦 = suc 1o)
35	34	fveq2d 6838	. . . . . . . . . . . 12 ⊢ (𝑦 = 1o → (𝑅1‘suc 𝑦) = (𝑅1‘suc 1o))
36		df-1o 8397	. . . . . . . . . . . . . . . . 17 ⊢ 1o = suc ∅
37	36	fveq2i 6837	. . . . . . . . . . . . . . . 16 ⊢ (𝑅1‘1o) = (𝑅1‘suc ∅)
38		0elon 6372	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ On
39		r1suc 9682	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅))
40	38, 39	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)
41		r10 9680	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅1‘∅) = ∅
42	41	pweqi 4570	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 (𝑅1‘∅) = 𝒫 ∅
43	37, 40, 42	3eqtri 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑅1‘1o) = 𝒫 ∅
44	43	pweqi 4570	. . . . . . . . . . . . . 14 ⊢ 𝒫 (𝑅1‘1o) = 𝒫 𝒫 ∅
45		pw0 4768	. . . . . . . . . . . . . . 15 ⊢ 𝒫 ∅ = {∅}
46	45	pweqi 4570	. . . . . . . . . . . . . 14 ⊢ 𝒫 𝒫 ∅ = 𝒫 {∅}
47		pwpw0 4769	. . . . . . . . . . . . . 14 ⊢ 𝒫 {∅} = {∅, {∅}}
48	44, 46, 47	3eqtrri 2764	. . . . . . . . . . . . 13 ⊢ {∅, {∅}} = 𝒫 (𝑅1‘1o)
49		1on 8409	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
50		r1suc 9682	. . . . . . . . . . . . . 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 2762	. . . . . . . . . . . 12 ⊢ {∅, {∅}} = (𝑅1‘suc 1o)
53	35, 52	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑦 = 1o → (𝑅1‘suc 𝑦) = {∅, {∅}})
54	53	eleq2d 2822	. . . . . . . . . 10 ⊢ (𝑦 = 1o → (𝑥 ∈ (𝑅1‘suc 𝑦) ↔ 𝑥 ∈ {∅, {∅}}))
55	54	elrab 3646	. . . . . . . . 9 ⊢ (1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
56	33, 55	sylib 218	. . . . . . . 8 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
57	12	elpr 4605	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
58		df-ne 2933	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
59		orel1 888	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
60	58, 59	sylbi 217	. . . . . . . . . . 11 ⊢ (𝑥 ≠ ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
61		df1o2 8404	. . . . . . . . . . . . 13 ⊢ 1o = {∅}
62		eqeq2 2748	. . . . . . . . . . . . 13 ⊢ (𝑥 = {∅} → (1o = 𝑥 ↔ 1o = {∅}))
63	61, 62	mpbiri 258	. . . . . . . . . . . 12 ⊢ (𝑥 = {∅} → 1o = 𝑥)
64	63	eqcomd 2742	. . . . . . . . . . 11 ⊢ (𝑥 = {∅} → 𝑥 = 1o)
65	60, 64	syl6com 37	. . . . . . . . . 10 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
66	57, 65	sylbi 217	. . . . . . . . 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 217	. . . . . 6 ⊢ ((rank‘𝑥) = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
70	15, 69	mpd 15	. . . . 5 ⊢ ((rank‘𝑥) = 1o → 𝑥 = 1o)
71	9, 70	vtoclg 3511	. . . 4 ⊢ (𝐴 ∈ V → ((rank‘𝐴) = 1o → 𝐴 = 1o))
72	6, 71	mpcom 38	. . 3 ⊢ ((rank‘𝐴) = 1o → 𝐴 = 1o)
73		fveq2 6834	. . . 4 ⊢ (𝐴 = 1o → (rank‘𝐴) = (rank‘1o))
74		r111 9687	. . . . . . 7 ⊢ 𝑅1:On–1-1→V
75		f1dm 6734	. . . . . . 7 ⊢ (𝑅1:On–1-1→V → dom 𝑅1 = On)
76	74, 75	ax-mp 5	. . . . . 6 ⊢ dom 𝑅1 = On
77	49, 76	eleqtrri 2835	. . . . 5 ⊢ 1o ∈ dom 𝑅1
78		rankonid 9741	. . . . 5 ⊢ (1o ∈ dom 𝑅1 ↔ (rank‘1o) = 1o)
79	77, 78	mpbi 230	. . . 4 ⊢ (rank‘1o) = 1o
80	73, 79	eqtrdi 2787	. . 3 ⊢ (𝐴 = 1o → (rank‘𝐴) = 1o)
81	72, 80	impbii 209	. 2 ⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = 1o)
82	61	eqeq2i 2749	. 2 ⊢ (𝐴 = 1o ↔ 𝐴 = {∅})
83	81, 82	bitri 275	1 ⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = {∅})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  {crab 3399  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  {cpr 4582  ∩ cint 4902  dom cdm 5624  Oncon0 6317  suc csuc 6319  –1-1→wf1 6489  ‘cfv 6492  1_oc1o 8390  𝑅₁cr1 9674  rankcrnk 9675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-reg 9497  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-r1 9676  df-rank 9677

This theorem is referenced by: (None)