Theorem lfuhgr1v0e 26146

Description: A loop-free hypergraph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 2-Apr-2021.)

Hypotheses

Ref	Expression
lfuhgr1v0e.v	⊢ 𝑉 = (Vtx‘𝐺)
lfuhgr1v0e.i	⊢ 𝐼 = (iEdg‘𝐺)
lfuhgr1v0e.e	⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}

Assertion

Ref	Expression
lfuhgr1v0e	⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → (Edg‘𝐺) = ∅)

Distinct variable groups:   𝑥,𝐺   𝑥,𝑉

Allowed substitution hints:   𝐸(𝑥)   𝐼(𝑥)

Proof of Theorem lfuhgr1v0e

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lfuhgr1v0e.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
2	1	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → 𝐼 = (iEdg‘𝐺))
3	1	dmeqi 5325	. . . . . 6 ⊢ dom 𝐼 = dom (iEdg‘𝐺)
4	3	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → dom 𝐼 = dom (iEdg‘𝐺))
5		lfuhgr1v0e.e	. . . . . 6 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}
6		lfuhgr1v0e.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
7		fvex 6201	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) ∈ V
8	6, 7	eqeltri 2697	. . . . . . . . 9 ⊢ 𝑉 ∈ V
9		hash1snb 13207	. . . . . . . . 9 ⊢ (𝑉 ∈ V → ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣}))
10	8, 9	ax-mp 5	. . . . . . . 8 ⊢ ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})
11		pweq 4161	. . . . . . . . . . . 12 ⊢ (𝑉 = {𝑣} → 𝒫 𝑉 = 𝒫 {𝑣})
12	11	rabeqdv 3194	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑣} → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = {𝑥 ∈ 𝒫 {𝑣} ∣ 2 ≤ (#‘𝑥)})
13		2pos 11112	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
14		0re 10040	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
15		2re 11090	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
16	14, 15	ltnlei 10158	. . . . . . . . . . . . . . 15 ⊢ (0 < 2 ↔ ¬ 2 ≤ 0)
17	13, 16	mpbi 220	. . . . . . . . . . . . . 14 ⊢ ¬ 2 ≤ 0
18		1lt2 11194	. . . . . . . . . . . . . . 15 ⊢ 1 < 2
19		1re 10039	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
20	19, 15	ltnlei 10158	. . . . . . . . . . . . . . 15 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
21	18, 20	mpbi 220	. . . . . . . . . . . . . 14 ⊢ ¬ 2 ≤ 1
22		0ex 4790	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
23		snex 4908	. . . . . . . . . . . . . . 15 ⊢ {𝑣} ∈ V
24		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (#‘𝑥) = (#‘∅))
25		hash0 13158	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘∅) = 0
26	24, 25	syl6eq 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (#‘𝑥) = 0)
27	26	breq2d 4665	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (2 ≤ (#‘𝑥) ↔ 2 ≤ 0))
28	27	notbid 308	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → (¬ 2 ≤ (#‘𝑥) ↔ ¬ 2 ≤ 0))
29		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = {𝑣} → (#‘𝑥) = (#‘{𝑣}))
30		vex 3203	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
31		hashsng 13159	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ V → (#‘{𝑣}) = 1)
32	30, 31	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘{𝑣}) = 1
33	29, 32	syl6eq 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = {𝑣} → (#‘𝑥) = 1)
34	33	breq2d 4665	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = {𝑣} → (2 ≤ (#‘𝑥) ↔ 2 ≤ 1))
35	34	notbid 308	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = {𝑣} → (¬ 2 ≤ (#‘𝑥) ↔ ¬ 2 ≤ 1))
36	22, 23, 28, 35	ralpr 4238	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥) ↔ (¬ 2 ≤ 0 ∧ ¬ 2 ≤ 1))
37	17, 21, 36	mpbir2an 955	. . . . . . . . . . . . 13 ⊢ ∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥)
38		pwsn 4428	. . . . . . . . . . . . . 14 ⊢ 𝒫 {𝑣} = {∅, {𝑣}}
39	38	raleqi 3142	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥) ↔ ∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥))
40	37, 39	mpbir 221	. . . . . . . . . . . 12 ⊢ ∀𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥)
41		rabeq0 3957	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ 𝒫 {𝑣} ∣ 2 ≤ (#‘𝑥)} = ∅ ↔ ∀𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥))
42	40, 41	mpbir 221	. . . . . . . . . . 11 ⊢ {𝑥 ∈ 𝒫 {𝑣} ∣ 2 ≤ (#‘𝑥)} = ∅
43	12, 42	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝑉 = {𝑣} → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅)
44	43	a1d 25	. . . . . . . . 9 ⊢ (𝑉 = {𝑣} → (𝐺 ∈ UHGraph → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅))
45	44	exlimiv 1858	. . . . . . . 8 ⊢ (∃𝑣 𝑉 = {𝑣} → (𝐺 ∈ UHGraph → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅))
46	10, 45	sylbi 207	. . . . . . 7 ⊢ ((#‘𝑉) = 1 → (𝐺 ∈ UHGraph → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅))
47	46	impcom 446	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅)
48	5, 47	syl5eq 2668	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → 𝐸 = ∅)
49	2, 4, 48	feq123d 6034	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → (𝐼:dom 𝐼⟶𝐸 ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅))
50	49	biimp3a 1432	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅)
51		f00 6087	. . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅))
52	51	simplbi 476	. . 3 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ → (iEdg‘𝐺) = ∅)
53	50, 52	syl 17	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → (iEdg‘𝐺) = ∅)
54		uhgriedg0edg0 26022	. . 3 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
55	54	3ad2ant1 1082	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
56	53, 55	mpbird 247	1 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → (Edg‘𝐺) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200  ∅c0 3915  𝒫 cpw 4158  {csn 4177  {cpr 4179   class class class wbr 4653  dom cdm 5114  ⟶wf 5884  ‘cfv 5888  0cc0 9936  1c1 9937   < clt 10074   ≤ cle 10075  2c2 11070  #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-uhgr 25953

This theorem is referenced by:  usgr1vr  26147  vtxdlfuhgr1v  26375