Theorem lfuhgr1v0e 26073

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 5295	. . . . . 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 6168	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) ∈ V
8	6, 7	eqeltri 2694	. . . . . . . . 9 ⊢ 𝑉 ∈ V
9		hash1snb 13163	. . . . . . . . 9 ⊢ (𝑉 ∈ V → ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣}))
10	8, 9	ax-mp 5	. . . . . . . 8 ⊢ ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})
11		pweq 4139	. . . . . . . . . . . 12 ⊢ (𝑉 = {𝑣} → 𝒫 𝑉 = 𝒫 {𝑣})
12	11	rabeqdv 3184	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑣} → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = {𝑥 ∈ 𝒫 {𝑣} ∣ 2 ≤ (#‘𝑥)})
13		2pos 11072	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
14		0re 10000	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
15		2re 11050	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
16	14, 15	ltnlei 10118	. . . . . . . . . . . . . . 15 ⊢ (0 < 2 ↔ ¬ 2 ≤ 0)
17	13, 16	mpbi 220	. . . . . . . . . . . . . 14 ⊢ ¬ 2 ≤ 0
18		1lt2 11154	. . . . . . . . . . . . . . 15 ⊢ 1 < 2
19		1re 9999	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
20	19, 15	ltnlei 10118	. . . . . . . . . . . . . . 15 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
21	18, 20	mpbi 220	. . . . . . . . . . . . . 14 ⊢ ¬ 2 ≤ 1
22		0ex 4760	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
23		snex 4879	. . . . . . . . . . . . . . 15 ⊢ {𝑣} ∈ V
24		fveq2 6158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (#‘𝑥) = (#‘∅))
25		hash0 13114	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘∅) = 0
26	24, 25	syl6eq 2671	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (#‘𝑥) = 0)
27	26	breq2d 4635	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (2 ≤ (#‘𝑥) ↔ 2 ≤ 0))
28	27	notbid 308	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → (¬ 2 ≤ (#‘𝑥) ↔ ¬ 2 ≤ 0))
29		fveq2 6158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = {𝑣} → (#‘𝑥) = (#‘{𝑣}))
30		vex 3193	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
31		hashsng 13115	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ V → (#‘{𝑣}) = 1)
32	30, 31	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘{𝑣}) = 1
33	29, 32	syl6eq 2671	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = {𝑣} → (#‘𝑥) = 1)
34	33	breq2d 4635	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = {𝑣} → (2 ≤ (#‘𝑥) ↔ 2 ≤ 1))
35	34	notbid 308	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = {𝑣} → (¬ 2 ≤ (#‘𝑥) ↔ ¬ 2 ≤ 1))
36	22, 23, 28, 35	ralpr 4216	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥) ↔ (¬ 2 ≤ 0 ∧ ¬ 2 ≤ 1))
37	17, 21, 36	mpbir2an 954	. . . . . . . . . . . . 13 ⊢ ∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥)
38		pwsn 4403	. . . . . . . . . . . . . 14 ⊢ 𝒫 {𝑣} = {∅, {𝑣}}
39	38	raleqi 3135	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥) ↔ ∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥))
40	37, 39	mpbir 221	. . . . . . . . . . . 12 ⊢ ∀𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥)
41		rabeq0 3937	. . . . . . . . . . . 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 2671	. . . . . . . . . 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 1855	. . . . . . . 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 2667	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → 𝐸 = ∅)
49	2, 4, 48	feq123d 6001	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → (𝐼:dom 𝐼⟶𝐸 ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅))
50	49	biimp3a 1429	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅)
51		f00 6054	. . . 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 25951	. . 3 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
55	54	3ad2ant1 1080	. 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 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∀wral 2908  {crab 2912  Vcvv 3190  ∅c0 3897  𝒫 cpw 4136  {csn 4155  {cpr 4157   class class class wbr 4623  dom cdm 5084  ⟶wf 5853  ‘cfv 5857  0cc0 9896  1c1 9897   < clt 10034   ≤ cle 10035  2c2 11030  #chash 13073  Vtxcvtx 25808  iEdgciedg 25809  Edgcedg 25873   UHGraph cuhgr 25881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-hash 13074  df-edg 25874  df-uhgr 25883

This theorem is referenced by:  usgr1vr  26074  vtxdlfuhgr1v  26295