Theorem rankeq1o 31448

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 7462	. . . . . . 7 ⊢ 1𝑜 ≠ ∅
2		neeq1 2844	. . . . . . 7 ⊢ ((rank‘𝐴) = 1𝑜 → ((rank‘𝐴) ≠ ∅ ↔ 1𝑜 ≠ ∅))
3	1, 2	mpbiri 247	. . . . . 6 ⊢ ((rank‘𝐴) = 1𝑜 → (rank‘𝐴) ≠ ∅)
4	3	neneqd 2787	. . . . 5 ⊢ ((rank‘𝐴) = 1𝑜 → ¬ (rank‘𝐴) = ∅)
5		fvprc 6097	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅)
6	4, 5	nsyl2 141	. . . 4 ⊢ ((rank‘𝐴) = 1𝑜 → 𝐴 ∈ V)
7		fveq2 6103	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
8	7	eqeq1d 2612	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 1𝑜 ↔ (rank‘𝐴) = 1𝑜))
9		eqeq1 2614	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 1𝑜 ↔ 𝐴 = 1𝑜))
10	8, 9	imbi12d 333	. . . . 5 ⊢ (𝑥 = 𝐴 → (((rank‘𝑥) = 1𝑜 → 𝑥 = 1𝑜) ↔ ((rank‘𝐴) = 1𝑜 → 𝐴 = 1𝑜)))
11		neeq1 2844	. . . . . . . 8 ⊢ ((rank‘𝑥) = 1𝑜 → ((rank‘𝑥) ≠ ∅ ↔ 1𝑜 ≠ ∅))
12	1, 11	mpbiri 247	. . . . . . 7 ⊢ ((rank‘𝑥) = 1𝑜 → (rank‘𝑥) ≠ ∅)
13		vex 3176	. . . . . . . . 9 ⊢ 𝑥 ∈ V
14	13	rankeq0 8607	. . . . . . . 8 ⊢ (𝑥 = ∅ ↔ (rank‘𝑥) = ∅)
15	14	necon3bii 2834	. . . . . . 7 ⊢ (𝑥 ≠ ∅ ↔ (rank‘𝑥) ≠ ∅)
16	12, 15	sylibr 223	. . . . . 6 ⊢ ((rank‘𝑥) = 1𝑜 → 𝑥 ≠ ∅)
17	13	rankval 8562	. . . . . . . 8 ⊢ (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}
18	17	eqeq1i 2615	. . . . . . 7 ⊢ ((rank‘𝑥) = 1𝑜 ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜)
19		ssrab2 3650	. . . . . . . . . . 11 ⊢ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
20		elirr 8388	. . . . . . . . . . . . . 14 ⊢ ¬ 1𝑜 ∈ 1𝑜
21		df1o2 7459	. . . . . . . . . . . . . . . 16 ⊢ 1𝑜 = {∅}
22		p0ex 4779	. . . . . . . . . . . . . . . 16 ⊢ {∅} ∈ V
23	21, 22	eqeltri 2684	. . . . . . . . . . . . . . 15 ⊢ 1𝑜 ∈ V
24		id 22	. . . . . . . . . . . . . . 15 ⊢ (V = 1𝑜 → V = 1𝑜)
25	23, 24	syl5eleq 2694	. . . . . . . . . . . . . 14 ⊢ (V = 1𝑜 → 1𝑜 ∈ 1𝑜)
26	20, 25	mto 187	. . . . . . . . . . . . 13 ⊢ ¬ V = 1𝑜
27		inteq 4413	. . . . . . . . . . . . . . 15 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∩ ∅)
28		int0 4425	. . . . . . . . . . . . . . 15 ⊢ ∩ ∅ = V
29	27, 28	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = V)
30	29	eqeq1d 2612	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 ↔ V = 1𝑜))
31	26, 30	mtbiri 316	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ¬ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜)
32	31	necon2ai 2811	. . . . . . . . . . 11 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
33		onint 6887	. . . . . . . . . . 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 694	. . . . . . . . . 10 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
35		eleq1 2676	. . . . . . . . . 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 221	. . . . . . . . 9 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → 1𝑜 ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
37		suceq 5707	. . . . . . . . . . . . 13 ⊢ (𝑦 = 1𝑜 → suc 𝑦 = suc 1𝑜)
38	37	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝑦 = 1𝑜 → (𝑅1‘suc 𝑦) = (𝑅1‘suc 1𝑜))
39		df-1o 7447	. . . . . . . . . . . . . . . . 17 ⊢ 1𝑜 = suc ∅
40	39	fveq2i 6106	. . . . . . . . . . . . . . . 16 ⊢ (𝑅1‘1𝑜) = (𝑅1‘suc ∅)
41		0elon 5695	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ On
42		r1suc 8516	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅))
43	41, 42	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)
44		r10 8514	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅1‘∅) = ∅
45	44	pweqi 4112	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 (𝑅1‘∅) = 𝒫 ∅
46	40, 43, 45	3eqtri 2636	. . . . . . . . . . . . . . 15 ⊢ (𝑅1‘1𝑜) = 𝒫 ∅
47	46	pweqi 4112	. . . . . . . . . . . . . 14 ⊢ 𝒫 (𝑅1‘1𝑜) = 𝒫 𝒫 ∅
48		pw0 4283	. . . . . . . . . . . . . . 15 ⊢ 𝒫 ∅ = {∅}
49	48	pweqi 4112	. . . . . . . . . . . . . 14 ⊢ 𝒫 𝒫 ∅ = 𝒫 {∅}
50		pwpw0 4284	. . . . . . . . . . . . . 14 ⊢ 𝒫 {∅} = {∅, {∅}}
51	47, 49, 50	3eqtrri 2637	. . . . . . . . . . . . 13 ⊢ {∅, {∅}} = 𝒫 (𝑅1‘1𝑜)
52		1on 7454	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ On
53		r1suc 8516	. . . . . . . . . . . . . 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 2635	. . . . . . . . . . . 12 ⊢ {∅, {∅}} = (𝑅1‘suc 1𝑜)
56	38, 55	syl6eqr 2662	. . . . . . . . . . 11 ⊢ (𝑦 = 1𝑜 → (𝑅1‘suc 𝑦) = {∅, {∅}})
57	56	eleq2d 2673	. . . . . . . . . 10 ⊢ (𝑦 = 1𝑜 → (𝑥 ∈ (𝑅1‘suc 𝑦) ↔ 𝑥 ∈ {∅, {∅}}))
58	57	elrab 3331	. . . . . . . . 9 ⊢ (1𝑜 ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ (1𝑜 ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
59	36, 58	sylib 207	. . . . . . . 8 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → (1𝑜 ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
60	13	elpr 4146	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
61		df-ne 2782	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
62		orel1 396	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
63	61, 62	sylbi 206	. . . . . . . . . . 11 ⊢ (𝑥 ≠ ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
64		eqeq2 2621	. . . . . . . . . . . . 13 ⊢ (𝑥 = {∅} → (1𝑜 = 𝑥 ↔ 1𝑜 = {∅}))
65	21, 64	mpbiri 247	. . . . . . . . . . . 12 ⊢ (𝑥 = {∅} → 1𝑜 = 𝑥)
66	65	eqcomd 2616	. . . . . . . . . . 11 ⊢ (𝑥 = {∅} → 𝑥 = 1𝑜)
67	63, 66	syl6com 36	. . . . . . . . . 10 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 ≠ ∅ → 𝑥 = 1𝑜))
68	60, 67	sylbi 206	. . . . . . . . 9 ⊢ (𝑥 ∈ {∅, {∅}} → (𝑥 ≠ ∅ → 𝑥 = 1𝑜))
69	68	adantl 481	. . . . . . . 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 206	. . . . . 6 ⊢ ((rank‘𝑥) = 1𝑜 → (𝑥 ≠ ∅ → 𝑥 = 1𝑜))
72	16, 71	mpd 15	. . . . 5 ⊢ ((rank‘𝑥) = 1𝑜 → 𝑥 = 1𝑜)
73	10, 72	vtoclg 3239	. . . 4 ⊢ (𝐴 ∈ V → ((rank‘𝐴) = 1𝑜 → 𝐴 = 1𝑜))
74	6, 73	mpcom 37	. . 3 ⊢ ((rank‘𝐴) = 1𝑜 → 𝐴 = 1𝑜)
75		fveq2 6103	. . . 4 ⊢ (𝐴 = 1𝑜 → (rank‘𝐴) = (rank‘1𝑜))
76		r111 8521	. . . . . . 7 ⊢ 𝑅1:On–1-1→V
77		f1dm 6018	. . . . . . 7 ⊢ (𝑅1:On–1-1→V → dom 𝑅1 = On)
78	76, 77	ax-mp 5	. . . . . 6 ⊢ dom 𝑅1 = On
79	52, 78	eleqtrri 2687	. . . . 5 ⊢ 1𝑜 ∈ dom 𝑅1
80		rankonid 8575	. . . . 5 ⊢ (1𝑜 ∈ dom 𝑅1 ↔ (rank‘1𝑜) = 1𝑜)
81	79, 80	mpbi 219	. . . 4 ⊢ (rank‘1𝑜) = 1𝑜
82	75, 81	syl6eq 2660	. . 3 ⊢ (𝐴 = 1𝑜 → (rank‘𝐴) = 1𝑜)
83	74, 82	impbii 198	. 2 ⊢ ((rank‘𝐴) = 1𝑜 ↔ 𝐴 = 1𝑜)
84	21	eqeq2i 2622	. 2 ⊢ (𝐴 = 1𝑜 ↔ 𝐴 = {∅})
85	83, 84	bitri 263	1 ⊢ ((rank‘𝐴) = 1𝑜 ↔ 𝐴 = {∅})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  ∩ cint 4410  dom cdm 5038  Oncon0 5640  suc csuc 5642  –1-1→wf1 5801  ‘cfv 5804  1_𝑜c1o 7440  𝑅₁cr1 8508  rankcrnk 8509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-r1 8510  df-rank 8511

This theorem is referenced by: (None)