Theorem rgrusgrprc 29607

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 5532	. . . . 5 ⊢ (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣∃𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
2		f0bi 6791	. . . . . . . . . 10 ⊢ (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
3		opeq2 4874	. . . . . . . . . . . 12 ⊢ (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ = ⟨𝑣, ∅⟩)
4		usgr0eop 29263	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ V → ⟨𝑣, ∅⟩ ∈ USGraph)
5	4	elv 3485	. . . . . . . . . . . 12 ⊢ ⟨𝑣, ∅⟩ ∈ USGraph
6	3, 5	eqeltrdi 2849	. . . . . . . . . . 11 ⊢ (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph)
7		vex 3484	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
8		vex 3484	. . . . . . . . . . . . 13 ⊢ 𝑒 ∈ V
9	7, 8	opiedgfvi 29027	. . . . . . . . . . . 12 ⊢ (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
10		id 22	. . . . . . . . . . . 12 ⊢ (𝑒 = ∅ → 𝑒 = ∅)
11	9, 10	eqtrid 2789	. . . . . . . . . . 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 2829	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → (𝑝 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph))
16		fveqeq2 6915	. . . . . . . . . 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 6915	. . . . . . . 8 ⊢ (𝑔 = 𝑝 → ((iEdg‘𝑔) = ∅ ↔ (iEdg‘𝑝) = ∅))
21	20	elrab 3692	. . . . . . 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 3987	. . 3 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
26		eqid 2737	. . . 4 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
27	26	griedg0prc 29281	. . 3 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V
28		prcssprc 5327	. . 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 1089	. . . . . . . 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 12605	. . . . . . 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 2737	. . . . . . . 8 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
37		eqid 2737	. . . . . . . 8 ⊢ (VtxDeg‘𝑔) = (VtxDeg‘𝑔)
38	36, 37	isrusgr0 29584	. . . . . . 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 3440	. . . 4 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0}
42		usgr0edg0rusgr 29593	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (Edg‘𝑔) = ∅))
43		usgruhgr 29203	. . . . . . 7 ⊢ (𝑔 ∈ USGraph → 𝑔 ∈ UHGraph)
44		uhgriedg0edg0 29144	. . . . . . 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 3440	. . . 4 ⊢ {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
48	41, 47	eqtri 2765	. . 3 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
49		neleq1 3052	. . 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 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ∉ wnel 3046  ∀wral 3061  {crab 3436  Vcvv 3480   ⊆ wss 3951  ∅c0 4333  ⟨cop 4632   class class class wbr 5143  {copab 5205  ⟶wf 6557  ‘cfv 6561  0cc0 11155  ℕ₀^*cxnn0 12599  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  UHGraphcuhgr 29073  USGraphcusgr 29166  VtxDegcvtxdg 29483   RegUSGraph crusgr 29574

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-xadd 13155  df-fz 13548  df-hash 14370  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-upgr 29099  df-uspgr 29167  df-usgr 29168  df-vtxdg 29484  df-rgr 29575  df-rusgr 29576

This theorem is referenced by:  rusgrprc  29608