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Theorem lfuhgr1v0e 26127
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 𝐼𝐸) → (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.
StepHypRef Expression
1 lfuhgr1v0e.i . . . . . 6 𝐼 = (iEdg‘𝐺)
21a1i 11 . . . . 5 ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → 𝐼 = (iEdg‘𝐺))
31dmeqi 5314 . . . . . 6 dom 𝐼 = dom (iEdg‘𝐺)
43a1i 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 6188 . . . . . . . . . 10 (Vtx‘𝐺) ∈ V
86, 7eqeltri 2695 . . . . . . . . 9 𝑉 ∈ V
9 hash1snb 13190 . . . . . . . . 9 (𝑉 ∈ V → ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣}))
108, 9ax-mp 5 . . . . . . . 8 ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})
11 pweq 4152 . . . . . . . . . . . 12 (𝑉 = {𝑣} → 𝒫 𝑉 = 𝒫 {𝑣})
1211rabeqdv 3189 . . . . . . . . . . 11 (𝑉 = {𝑣} → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = {𝑥 ∈ 𝒫 {𝑣} ∣ 2 ≤ (#‘𝑥)})
13 2pos 11097 . . . . . . . . . . . . . . 15 0 < 2
14 0re 10025 . . . . . . . . . . . . . . . 16 0 ∈ ℝ
15 2re 11075 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
1614, 15ltnlei 10143 . . . . . . . . . . . . . . 15 (0 < 2 ↔ ¬ 2 ≤ 0)
1713, 16mpbi 220 . . . . . . . . . . . . . 14 ¬ 2 ≤ 0
18 1lt2 11179 . . . . . . . . . . . . . . 15 1 < 2
19 1re 10024 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
2019, 15ltnlei 10143 . . . . . . . . . . . . . . 15 (1 < 2 ↔ ¬ 2 ≤ 1)
2118, 20mpbi 220 . . . . . . . . . . . . . 14 ¬ 2 ≤ 1
22 0ex 4781 . . . . . . . . . . . . . . 15 ∅ ∈ V
23 snex 4899 . . . . . . . . . . . . . . 15 {𝑣} ∈ V
24 fveq2 6178 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → (#‘𝑥) = (#‘∅))
25 hash0 13141 . . . . . . . . . . . . . . . . . 18 (#‘∅) = 0
2624, 25syl6eq 2670 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → (#‘𝑥) = 0)
2726breq2d 4656 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (2 ≤ (#‘𝑥) ↔ 2 ≤ 0))
2827notbid 308 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (¬ 2 ≤ (#‘𝑥) ↔ ¬ 2 ≤ 0))
29 fveq2 6178 . . . . . . . . . . . . . . . . . 18 (𝑥 = {𝑣} → (#‘𝑥) = (#‘{𝑣}))
30 vex 3198 . . . . . . . . . . . . . . . . . . 19 𝑣 ∈ V
31 hashsng 13142 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ V → (#‘{𝑣}) = 1)
3230, 31ax-mp 5 . . . . . . . . . . . . . . . . . 18 (#‘{𝑣}) = 1
3329, 32syl6eq 2670 . . . . . . . . . . . . . . . . 17 (𝑥 = {𝑣} → (#‘𝑥) = 1)
3433breq2d 4656 . . . . . . . . . . . . . . . 16 (𝑥 = {𝑣} → (2 ≤ (#‘𝑥) ↔ 2 ≤ 1))
3534notbid 308 . . . . . . . . . . . . . . 15 (𝑥 = {𝑣} → (¬ 2 ≤ (#‘𝑥) ↔ ¬ 2 ≤ 1))
3622, 23, 28, 35ralpr 4229 . . . . . . . . . . . . . 14 (∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥) ↔ (¬ 2 ≤ 0 ∧ ¬ 2 ≤ 1))
3717, 21, 36mpbir2an 954 . . . . . . . . . . . . 13 𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥)
38 pwsn 4419 . . . . . . . . . . . . . 14 𝒫 {𝑣} = {∅, {𝑣}}
3938raleqi 3137 . . . . . . . . . . . . 13 (∀𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥) ↔ ∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥))
4037, 39mpbir 221 . . . . . . . . . . . 12 𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥)
41 rabeq0 3948 . . . . . . . . . . . 12 ({𝑥 ∈ 𝒫 {𝑣} ∣ 2 ≤ (#‘𝑥)} = ∅ ↔ ∀𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥))
4240, 41mpbir 221 . . . . . . . . . . 11 {𝑥 ∈ 𝒫 {𝑣} ∣ 2 ≤ (#‘𝑥)} = ∅
4312, 42syl6eq 2670 . . . . . . . . . 10 (𝑉 = {𝑣} → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅)
4443a1d 25 . . . . . . . . 9 (𝑉 = {𝑣} → (𝐺 ∈ UHGraph → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅))
4544exlimiv 1856 . . . . . . . 8 (∃𝑣 𝑉 = {𝑣} → (𝐺 ∈ UHGraph → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅))
4610, 45sylbi 207 . . . . . . 7 ((#‘𝑉) = 1 → (𝐺 ∈ UHGraph → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅))
4746impcom 446 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅)
485, 47syl5eq 2666 . . . . 5 ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → 𝐸 = ∅)
492, 4, 48feq123d 6021 . . . 4 ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → (𝐼:dom 𝐼𝐸 ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅))
5049biimp3a 1430 . . 3 ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼𝐸) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅)
51 f00 6074 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅))
5251simplbi 476 . . 3 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ → (iEdg‘𝐺) = ∅)
5350, 52syl 17 . 2 ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼𝐸) → (iEdg‘𝐺) = ∅)
54 uhgriedg0edg0 26003 . . 3 (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
55543ad2ant1 1080 . 2 ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼𝐸) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
5653, 55mpbird 247 1 ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼𝐸) → (Edg‘𝐺) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wex 1702  wcel 1988  wral 2909  {crab 2913  Vcvv 3195  c0 3907  𝒫 cpw 4149  {csn 4168  {cpr 4170   class class class wbr 4644  dom cdm 5104  wf 5872  cfv 5876  0cc0 9921  1c1 9922   < clt 10059  cle 10060  2c2 11055  #chash 13100  Vtxcvtx 25855  iEdgciedg 25856  Edgcedg 25920   UHGraph cuhgr 25932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-hash 13101  df-edg 25921  df-uhgr 25934
This theorem is referenced by:  usgr1vr  26128  vtxdlfuhgr1v  26356
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