Theorem rgrusgrprc 29844

Description: The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)

Assertion

Ref	Expression
rgrusgrprc	⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V

Distinct variable group:   𝑣,𝑔

Proof of Theorem rgrusgrprc

Dummy variables 𝑒 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elopab 5501	. . . . 5 ⊢ (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣∃𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
2		f0bi 6751	. . . . . . . . . 10 ⊢ (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
3		opeq2 4834	. . . . . . . . . . . 12 ⊢ (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ = ⟨𝑣, ∅⟩)
4		usgr0eop 29501	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ V → ⟨𝑣, ∅⟩ ∈ USGraph)
5	4	elv 3462	. . . . . . . . . . . 12 ⊢ ⟨𝑣, ∅⟩ ∈ USGraph
6	3, 5	eqeltrdi 2873	. . . . . . . . . . 11 ⊢ (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph)
7		vex 3461	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
8		vex 3461	. . . . . . . . . . . . 13 ⊢ 𝑒 ∈ V
9	7, 8	opiedgfvi 29265	. . . . . . . . . . . 12 ⊢ (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
10		id 23	. . . . . . . . . . . 12 ⊢ (𝑒 = ∅ → 𝑒 = ∅)
11	9, 10	eqtrid 2812	. . . . . . . . . . 11 ⊢ (𝑒 = ∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
12	6, 11	jca 520	. . . . . . . . . 10 ⊢ (𝑒 = ∅ → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
13	2, 12	sylbi 220	. . . . . . . . 9 ⊢ (𝑒:∅⟶∅ → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
14	13	adantl 486	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
15		eleq1 2853	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → (𝑝 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph))
16		fveqeq2 6880	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → ((iEdg‘𝑝) = ∅ ↔ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
17	15, 16	anbi12d 643	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → ((𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅) ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅)))
18	17	adantr 485	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ((𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅) ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅)))
19	14, 18	mpbird 260	. . . . . . 7 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅))
20		fveqeq2 6880	. . . . . . . 8 ⊢ (𝑔 = 𝑝 → ((iEdg‘𝑔) = ∅ ↔ (iEdg‘𝑝) = ∅))
21	20	elrab 3653	. . . . . . 7 ⊢ (𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ↔ (𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅))
22	19, 21	sylibr 237	. . . . . 6 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
23	22	exlimivv 1955	. . . . 5 ⊢ (∃𝑣∃𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
24	1, 23	sylbi 220	. . . 4 ⊢ (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
25	24	ssriv 3943	. . 3 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
26		eqid 2765	. . . 4 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
27	26	griedg0prc 29519	. . 3 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V
28		prcssprc 5287	. . 3 ⊢ (({⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∧ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V) → {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V)
29	25, 27, 28	mp2an 704	. 2 ⊢ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V
30		df-3an 1103	. . . . . . . 8 ⊢ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0) ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0))
31	30	bicomi 227	. . . . . . 7 ⊢ (((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0) ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0))
32	31	a1i 11	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0) ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
33		0xnn0 12571	. . . . . . 7 ⊢ 0 ∈ ℕ0*
34		ibar 537	. . . . . . 7 ⊢ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
35	33, 34	mpan2 703	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
36		eqid 2765	. . . . . . . 8 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
37		eqid 2765	. . . . . . . 8 ⊢ (VtxDeg‘𝑔) = (VtxDeg‘𝑔)
38	36, 37	isrusgr0 29821	. . . . . . 7 ⊢ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) → (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
39	33, 38	mpan2 703	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
40	32, 35, 39	3bitr4d 314	. . . . 5 ⊢ (𝑔 ∈ USGraph → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ 𝑔 RegUSGraph 0))
41	40	rabbiia 3421	. . . 4 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0}
42		usgr0edg0rusgr 29830	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (Edg‘𝑔) = ∅))
43		usgruhgr 29441	. . . . . . 7 ⊢ (𝑔 ∈ USGraph → 𝑔 ∈ UHGraph)
44		uhgriedg0edg0 29382	. . . . . . 7 ⊢ (𝑔 ∈ UHGraph → ((Edg‘𝑔) = ∅ ↔ (iEdg‘𝑔) = ∅))
45	43, 44	syl 18	. . . . . 6 ⊢ (𝑔 ∈ USGraph → ((Edg‘𝑔) = ∅ ↔ (iEdg‘𝑔) = ∅))
46	42, 45	bitrd 282	. . . . 5 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (iEdg‘𝑔) = ∅))
47	46	rabbiia 3421	. . . 4 ⊢ {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
48	41, 47	eqtri 2788	. . 3 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
49		neleq1 3070	. . 3 ⊢ ({𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} → ({𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V ↔ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V))
50	48, 49	ax-mp 5	. 2 ⊢ ({𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V ↔ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V)
51	29, 50	mpbir 234	1 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145   ∉ wnel 3064  ∀wral 3079  {crab 3417  Vcvv 3457   ⊆ wss 3907  ∅c0 4288  ⟨cop 4591   class class class wbr 5104  {copab 5166  ⟶wf 6521  ‘cfv 6525  0cc0 11088  ℕ₀^*cxnn0 12565  Vtxcvtx 29251  iEdgciedg 29252  Edgcedg 29302  UHGraphcuhgr 29311  USGraphcusgr 29404  VtxDegcvtxdg 29720   RegUSGraph crusgr 29811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12222  df-2 12291  df-n0 12493  df-xnn0 12566  df-z 12580  df-uz 12851  df-xadd 13126  df-fz 13524  df-hash 14355  df-iedg 29254  df-edg 29303  df-uhgr 29313  df-upgr 29337  df-uspgr 29405  df-usgr 29406  df-vtxdg 29721  df-rgr 29812  df-rusgr 29813

This theorem is referenced by:  rusgrprc  29845