Theorem rgrusgrprc 29523

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 5489	. . . . 5 ⊢ (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣∃𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
2		f0bi 6745	. . . . . . . . . 10 ⊢ (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
3		opeq2 4840	. . . . . . . . . . . 12 ⊢ (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ = ⟨𝑣, ∅⟩)
4		usgr0eop 29179	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ V → ⟨𝑣, ∅⟩ ∈ USGraph)
5	4	elv 3455	. . . . . . . . . . . 12 ⊢ ⟨𝑣, ∅⟩ ∈ USGraph
6	3, 5	eqeltrdi 2837	. . . . . . . . . . 11 ⊢ (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph)
7		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
8		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑒 ∈ V
9	7, 8	opiedgfvi 28943	. . . . . . . . . . . 12 ⊢ (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
10		id 22	. . . . . . . . . . . 12 ⊢ (𝑒 = ∅ → 𝑒 = ∅)
11	9, 10	eqtrid 2777	. . . . . . . . . . 11 ⊢ (𝑒 = ∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
12	6, 11	jca 511	. . . . . . . . . 10 ⊢ (𝑒 = ∅ → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
13	2, 12	sylbi 217	. . . . . . . . 9 ⊢ (𝑒:∅⟶∅ → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
14	13	adantl 481	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
15		eleq1 2817	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → (𝑝 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph))
16		fveqeq2 6869	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → ((iEdg‘𝑝) = ∅ ↔ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
17	15, 16	anbi12d 632	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → ((𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅) ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅)))
18	17	adantr 480	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ((𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅) ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅)))
19	14, 18	mpbird 257	. . . . . . 7 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅))
20		fveqeq2 6869	. . . . . . . 8 ⊢ (𝑔 = 𝑝 → ((iEdg‘𝑔) = ∅ ↔ (iEdg‘𝑝) = ∅))
21	20	elrab 3661	. . . . . . 7 ⊢ (𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ↔ (𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅))
22	19, 21	sylibr 234	. . . . . 6 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
23	22	exlimivv 1932	. . . . 5 ⊢ (∃𝑣∃𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
24	1, 23	sylbi 217	. . . 4 ⊢ (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
25	24	ssriv 3952	. . 3 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
26		eqid 2730	. . . 4 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
27	26	griedg0prc 29197	. . 3 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V
28		prcssprc 5284	. . 3 ⊢ (({⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∧ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V) → {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V)
29	25, 27, 28	mp2an 692	. 2 ⊢ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V
30		df-3an 1088	. . . . . . . 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 224	. . . . . . 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 12527	. . . . . . 7 ⊢ 0 ∈ ℕ0*
34		ibar 528	. . . . . . 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 691	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
36		eqid 2730	. . . . . . . 8 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
37		eqid 2730	. . . . . . . 8 ⊢ (VtxDeg‘𝑔) = (VtxDeg‘𝑔)
38	36, 37	isrusgr0 29500	. . . . . . 7 ⊢ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) → (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
39	33, 38	mpan2 691	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
40	32, 35, 39	3bitr4d 311	. . . . 5 ⊢ (𝑔 ∈ USGraph → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ 𝑔 RegUSGraph 0))
41	40	rabbiia 3412	. . . 4 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0}
42		usgr0edg0rusgr 29509	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (Edg‘𝑔) = ∅))
43		usgruhgr 29119	. . . . . . 7 ⊢ (𝑔 ∈ USGraph → 𝑔 ∈ UHGraph)
44		uhgriedg0edg0 29060	. . . . . . 7 ⊢ (𝑔 ∈ UHGraph → ((Edg‘𝑔) = ∅ ↔ (iEdg‘𝑔) = ∅))
45	43, 44	syl 17	. . . . . 6 ⊢ (𝑔 ∈ USGraph → ((Edg‘𝑔) = ∅ ↔ (iEdg‘𝑔) = ∅))
46	42, 45	bitrd 279	. . . . 5 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (iEdg‘𝑔) = ∅))
47	46	rabbiia 3412	. . . 4 ⊢ {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
48	41, 47	eqtri 2753	. . 3 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
49		neleq1 3036	. . 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 231	1 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ∉ wnel 3030  ∀wral 3045  {crab 3408  Vcvv 3450   ⊆ wss 3916  ∅c0 4298  ⟨cop 4597   class class class wbr 5109  {copab 5171  ⟶wf 6509  ‘cfv 6513  0cc0 11074  ℕ₀^*cxnn0 12521  Vtxcvtx 28929  iEdgciedg 28930  Edgcedg 28980  UHGraphcuhgr 28989  USGraphcusgr 29082  VtxDegcvtxdg 29399   RegUSGraph crusgr 29490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-card 9898  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-n0 12449  df-xnn0 12522  df-z 12536  df-uz 12800  df-xadd 13079  df-fz 13475  df-hash 14302  df-iedg 28932  df-edg 28981  df-uhgr 28991  df-upgr 29015  df-uspgr 29083  df-usgr 29084  df-vtxdg 29400  df-rgr 29491  df-rusgr 29492

This theorem is referenced by:  rusgrprc  29524