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Theorem usgravd00 26446
 Description: If every vertex in a simple graph has degree 0, there is no edge in the graph. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
Assertion
Ref Expression
usgravd00 (𝑉 USGrph 𝐸 → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → 𝐸 = ∅))
Distinct variable groups:   𝑣,𝐸   𝑣,𝑉

Proof of Theorem usgravd00
Dummy variables 𝑒 𝑝 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgraf0 25877 . 2 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
2 f1f 6014 . . 3 (𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → 𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
3 simpll 786 . . . . . 6 (((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0) → 𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
4 usgravd0nedg 26445 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑣𝑉) → (((𝑉 VDeg 𝐸)‘𝑣) = 0 → ¬ ∃𝑤𝑉 {𝑣, 𝑤} ∈ ran 𝐸))
54adantll 746 . . . . . . . . 9 (((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) ∧ 𝑣𝑉) → (((𝑉 VDeg 𝐸)‘𝑣) = 0 → ¬ ∃𝑤𝑉 {𝑣, 𝑤} ∈ ran 𝐸))
65ralimdva 2945 . . . . . . . 8 ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → ∀𝑣𝑉 ¬ ∃𝑤𝑉 {𝑣, 𝑤} ∈ ran 𝐸))
7 ralnex 2975 . . . . . . . . 9 (∀𝑣𝑉 ¬ ∃𝑤𝑉 {𝑣, 𝑤} ∈ ran 𝐸 ↔ ¬ ∃𝑣𝑉𝑤𝑉 {𝑣, 𝑤} ∈ ran 𝐸)
8 exprelprel 13126 . . . . . . . . . . 11 (∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝 ∈ ran 𝐸 → ∃𝑣𝑉𝑤𝑉 {𝑣, 𝑤} ∈ ran 𝐸)
98a1i 11 . . . . . . . . . 10 ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) → (∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝 ∈ ran 𝐸 → ∃𝑣𝑉𝑤𝑉 {𝑣, 𝑤} ∈ ran 𝐸))
109con3d 147 . . . . . . . . 9 ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) → (¬ ∃𝑣𝑉𝑤𝑉 {𝑣, 𝑤} ∈ ran 𝐸 → ¬ ∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝 ∈ ran 𝐸))
117, 10syl5bi 231 . . . . . . . 8 ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) → (∀𝑣𝑉 ¬ ∃𝑤𝑉 {𝑣, 𝑤} ∈ ran 𝐸 → ¬ ∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝 ∈ ran 𝐸))
126, 11syld 46 . . . . . . 7 ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → ¬ ∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝 ∈ ran 𝐸))
1312imp 444 . . . . . 6 (((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0) → ¬ ∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝 ∈ ran 𝐸)
14 f0rn0 6003 . . . . . 6 ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ ¬ ∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝 ∈ ran 𝐸) → dom 𝐸 = ∅)
153, 13, 14syl2anc 691 . . . . 5 (((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0) → dom 𝐸 = ∅)
16 frel 5963 . . . . . . 7 (𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → Rel 𝐸)
17 reldm0 5264 . . . . . . 7 (Rel 𝐸 → (𝐸 = ∅ ↔ dom 𝐸 = ∅))
1816, 17syl 17 . . . . . 6 (𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝐸 = ∅ ↔ dom 𝐸 = ∅))
1918ad2antrr 758 . . . . 5 (((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0) → (𝐸 = ∅ ↔ dom 𝐸 = ∅))
2015, 19mpbird 246 . . . 4 (((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0) → 𝐸 = ∅)
2120exp31 628 . . 3 (𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑉 USGrph 𝐸 → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → 𝐸 = ∅)))
222, 21syl 17 . 2 (𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑉 USGrph 𝐸 → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → 𝐸 = ∅)))
231, 22mpcom 37 1 (𝑉 USGrph 𝐸 → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → 𝐸 = ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  ∅c0 3874  𝒫 cpw 4108  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  Rel wrel 5043  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  0cc0 9815  2c2 10947  #chash 12979   USGrph cusg 25859   VDeg cvdg 26420 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-2o 7448  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-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-usgra 25862  df-vdgr 26421 This theorem is referenced by:  0eusgraiff0rgra  26466
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