Theorem lfuhgr1v0e 40480

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 5247	. . . . . 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 6113	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) ∈ V
8	6, 7	eqeltri 2684	. . . . . . . . 9 ⊢ 𝑉 ∈ V
9		hash1snb 13068	. . . . . . . . 9 ⊢ (𝑉 ∈ V → ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣}))
10	8, 9	ax-mp 5	. . . . . . . 8 ⊢ ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})
11		pweq 4111	. . . . . . . . . . . 12 ⊢ (𝑉 = {𝑣} → 𝒫 𝑉 = 𝒫 {𝑣})
12	11	rabeqdv 3167	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑣} → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = {𝑥 ∈ 𝒫 {𝑣} ∣ 2 ≤ (#‘𝑥)})
13		2pos 10989	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
14		0re 9919	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
15		2re 10967	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
16	14, 15	ltnlei 10037	. . . . . . . . . . . . . . 15 ⊢ (0 < 2 ↔ ¬ 2 ≤ 0)
17	13, 16	mpbi 219	. . . . . . . . . . . . . 14 ⊢ ¬ 2 ≤ 0
18		1lt2 11071	. . . . . . . . . . . . . . 15 ⊢ 1 < 2
19		1re 9918	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
20	19, 15	ltnlei 10037	. . . . . . . . . . . . . . 15 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
21	18, 20	mpbi 219	. . . . . . . . . . . . . 14 ⊢ ¬ 2 ≤ 1
22		0ex 4718	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
23		snex 4835	. . . . . . . . . . . . . . 15 ⊢ {𝑣} ∈ V
24		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → (#‘𝑥) = (#‘∅))
25		hash0 13019	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘∅) = 0
26	24, 25	syl6eq 2660	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (#‘𝑥) = 0)
27	26	breq2d 4595	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (2 ≤ (#‘𝑥) ↔ 2 ≤ 0))
28	27	notbid 307	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → (¬ 2 ≤ (#‘𝑥) ↔ ¬ 2 ≤ 0))
29		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = {𝑣} → (#‘𝑥) = (#‘{𝑣}))
30		vex 3176	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
31		hashsng 13020	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ V → (#‘{𝑣}) = 1)
32	30, 31	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘{𝑣}) = 1
33	29, 32	syl6eq 2660	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = {𝑣} → (#‘𝑥) = 1)
34	33	breq2d 4595	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = {𝑣} → (2 ≤ (#‘𝑥) ↔ 2 ≤ 1))
35	34	notbid 307	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = {𝑣} → (¬ 2 ≤ (#‘𝑥) ↔ ¬ 2 ≤ 1))
36	22, 23, 28, 35	ralpr 4185	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥) ↔ (¬ 2 ≤ 0 ∧ ¬ 2 ≤ 1))
37	17, 21, 36	mpbir2an 957	. . . . . . . . . . . . 13 ⊢ ∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥)
38		pwsn 4366	. . . . . . . . . . . . . 14 ⊢ 𝒫 {𝑣} = {∅, {𝑣}}
39	38	raleqi 3119	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥) ↔ ∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥))
40	37, 39	mpbir 220	. . . . . . . . . . . 12 ⊢ ∀𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥)
41		rabeq0 3911	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ 𝒫 {𝑣} ∣ 2 ≤ (#‘𝑥)} = ∅ ↔ ∀𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥))
42	40, 41	mpbir 220	. . . . . . . . . . 11 ⊢ {𝑥 ∈ 𝒫 {𝑣} ∣ 2 ≤ (#‘𝑥)} = ∅
43	12, 42	syl6eq 2660	. . . . . . . . . 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 1845	. . . . . . . 8 ⊢ (∃𝑣 𝑉 = {𝑣} → (𝐺 ∈ UHGraph → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅))
46	10, 45	sylbi 206	. . . . . . 7 ⊢ ((#‘𝑉) = 1 → (𝐺 ∈ UHGraph → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅))
47	46	impcom 445	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅)
48	5, 47	syl5eq 2656	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → 𝐸 = ∅)
49	2, 4, 48	feq123d 5947	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → (𝐼:dom 𝐼⟶𝐸 ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅))
50	49	biimp3a 1424	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅)
51		f00 6000	. . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅))
52	51	simplbi 475	. . 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 25801	. . 3 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
55	54	3ad2ant1 1075	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
56	53, 55	mpbird 246	1 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → (Edg‘𝐺) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  0cc0 9815  1c1 9816   < clt 9953   ≤ cle 9954  2c2 10947  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722  Edgcedga 25792

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-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

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-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  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-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-uhgr 25724  df-edga 25793

This theorem is referenced by:  usgr1vr  40481  vtxdlfuhgr1v  40694