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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 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 5295 . . . . . 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 6168 . . . . . . . . . 10 (Vtx‘𝐺) ∈ V
86, 7eqeltri 2694 . . . . . . . . 9 𝑉 ∈ V
9 hash1snb 13163 . . . . . . . . 9 (𝑉 ∈ V → ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣}))
108, 9ax-mp 5 . . . . . . . 8 ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})
11 pweq 4139 . . . . . . . . . . . 12 (𝑉 = {𝑣} → 𝒫 𝑉 = 𝒫 {𝑣})
1211rabeqdv 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
1614, 15ltnlei 10118 . . . . . . . . . . . . . . 15 (0 < 2 ↔ ¬ 2 ≤ 0)
1713, 16mpbi 220 . . . . . . . . . . . . . 14 ¬ 2 ≤ 0
18 1lt2 11154 . . . . . . . . . . . . . . 15 1 < 2
19 1re 9999 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
2019, 15ltnlei 10118 . . . . . . . . . . . . . . 15 (1 < 2 ↔ ¬ 2 ≤ 1)
2118, 20mpbi 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
2624, 25syl6eq 2671 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → (#‘𝑥) = 0)
2726breq2d 4635 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (2 ≤ (#‘𝑥) ↔ 2 ≤ 0))
2827notbid 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)
3230, 31ax-mp 5 . . . . . . . . . . . . . . . . . 18 (#‘{𝑣}) = 1
3329, 32syl6eq 2671 . . . . . . . . . . . . . . . . 17 (𝑥 = {𝑣} → (#‘𝑥) = 1)
3433breq2d 4635 . . . . . . . . . . . . . . . 16 (𝑥 = {𝑣} → (2 ≤ (#‘𝑥) ↔ 2 ≤ 1))
3534notbid 308 . . . . . . . . . . . . . . 15 (𝑥 = {𝑣} → (¬ 2 ≤ (#‘𝑥) ↔ ¬ 2 ≤ 1))
3622, 23, 28, 35ralpr 4216 . . . . . . . . . . . . . 14 (∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥) ↔ (¬ 2 ≤ 0 ∧ ¬ 2 ≤ 1))
3717, 21, 36mpbir2an 954 . . . . . . . . . . . . 13 𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥)
38 pwsn 4403 . . . . . . . . . . . . . 14 𝒫 {𝑣} = {∅, {𝑣}}
3938raleqi 3135 . . . . . . . . . . . . 13 (∀𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥) ↔ ∀𝑥 ∈ {∅, {𝑣}} ¬ 2 ≤ (#‘𝑥))
4037, 39mpbir 221 . . . . . . . . . . . 12 𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥)
41 rabeq0 3937 . . . . . . . . . . . 12 ({𝑥 ∈ 𝒫 {𝑣} ∣ 2 ≤ (#‘𝑥)} = ∅ ↔ ∀𝑥 ∈ 𝒫 {𝑣} ¬ 2 ≤ (#‘𝑥))
4240, 41mpbir 221 . . . . . . . . . . 11 {𝑥 ∈ 𝒫 {𝑣} ∣ 2 ≤ (#‘𝑥)} = ∅
4312, 42syl6eq 2671 . . . . . . . . . 10 (𝑉 = {𝑣} → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅)
4443a1d 25 . . . . . . . . 9 (𝑉 = {𝑣} → (𝐺 ∈ UHGraph → {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = ∅))
4544exlimiv 1855 . . . . . . . 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 2667 . . . . 5 ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → 𝐸 = ∅)
492, 4, 48feq123d 6001 . . . 4 ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1) → (𝐼:dom 𝐼𝐸 ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅))
5049biimp3a 1429 . . 3 ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼𝐸) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅)
51 f00 6054 . . . 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 25951 . . 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 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
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