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 → (∀𝑣 ∈ 𝑉 ((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.

Step	Hyp	Ref	Expression
1		usgraf0 25877	. 2 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
2		f1f 6014	. . 3 ⊢ (𝐸:dom 𝐸–1-1→{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) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0) → 𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
4		usgravd0nedg 26445	. . . . . . . . . 10 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑣 ∈ 𝑉) → (((𝑉 VDeg 𝐸)‘𝑣) = 0 → ¬ ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ ran 𝐸))
5	4	adantll 746	. . . . . . . . 9 ⊢ (((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) ∧ 𝑣 ∈ 𝑉) → (((𝑉 VDeg 𝐸)‘𝑣) = 0 → ¬ ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ ran 𝐸))
6	5	ralimdva 2945	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → ∀𝑣 ∈ 𝑉 ¬ ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ ran 𝐸))
7		ralnex 2975	. . . . . . . . 9 ⊢ (∀𝑣 ∈ 𝑉 ¬ ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ ran 𝐸 ↔ ¬ ∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ ran 𝐸)
8		exprelprel 13126	. . . . . . . . . . 11 ⊢ (∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝 ∈ ran 𝐸 → ∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ ran 𝐸)
9	8	a1i 11	. . . . . . . . . 10 ⊢ ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) → (∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝 ∈ ran 𝐸 → ∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ ran 𝐸))
10	9	con3d 147	. . . . . . . . 9 ⊢ ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) → (¬ ∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ ran 𝐸 → ¬ ∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝 ∈ ran 𝐸))
11	7, 10	syl5bi 231	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) → (∀𝑣 ∈ 𝑉 ¬ ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ ran 𝐸 → ¬ ∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝 ∈ ran 𝐸))
12	6, 11	syld 46	. . . . . . 7 ⊢ ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → ¬ ∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝 ∈ ran 𝐸))
13	12	imp 444	. . . . . 6 ⊢ (((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) ∧ ∀𝑣 ∈ 𝑉 ((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 = ∅)
15	3, 13, 14	syl2anc 691	. . . . 5 ⊢ (((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) ∧ ∀𝑣 ∈ 𝑉 ((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 = ∅))
18	16, 17	syl 17	. . . . . 6 ⊢ (𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝐸 = ∅ ↔ dom 𝐸 = ∅))
19	18	ad2antrr 758	. . . . 5 ⊢ (((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0) → (𝐸 = ∅ ↔ dom 𝐸 = ∅))
20	15, 19	mpbird 246	. . . 4 ⊢ (((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑉 USGrph 𝐸) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0) → 𝐸 = ∅)
21	20	exp31 628	. . 3 ⊢ (𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑉 USGrph 𝐸 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → 𝐸 = ∅)))
22	2, 21	syl 17	. 2 ⊢ (𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑉 USGrph 𝐸 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → 𝐸 = ∅)))
23	1, 22	mpcom 37	1 ⊢ (𝑉 USGrph 𝐸 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → 𝐸 = ∅))

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