Theorem rgrusgrprc 26485

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 4983	. . . . 5 ⊢ (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣∃𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
2		f0bi 6088	. . . . . . . . . 10 ⊢ (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
3		opeq2 4403	. . . . . . . . . . . 12 ⊢ (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ = ⟨𝑣, ∅⟩)
4		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
5		usgr0eop 26138	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ V → ⟨𝑣, ∅⟩ ∈ USGraph )
6	4, 5	ax-mp 5	. . . . . . . . . . . 12 ⊢ ⟨𝑣, ∅⟩ ∈ USGraph
7	3, 6	syl6eqel 2709	. . . . . . . . . . 11 ⊢ (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph )
8		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑒 ∈ V
9		opiedgfv 25887	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒)
10	4, 8, 9	mp2an 708	. . . . . . . . . . . 12 ⊢ (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
11		id 22	. . . . . . . . . . . 12 ⊢ (𝑒 = ∅ → 𝑒 = ∅)
12	10, 11	syl5eq 2668	. . . . . . . . . . 11 ⊢ (𝑒 = ∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
13	7, 12	jca 554	. . . . . . . . . 10 ⊢ (𝑒 = ∅ → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
14	2, 13	sylbi 207	. . . . . . . . 9 ⊢ (𝑒:∅⟶∅ → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
15	14	adantl 482	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
16		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → (𝑝 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph ))
17		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → (iEdg‘𝑝) = (iEdg‘⟨𝑣, 𝑒⟩))
18	17	eqeq1d 2624	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → ((iEdg‘𝑝) = ∅ ↔ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
19	16, 18	anbi12d 747	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → ((𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅) ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅)))
20	19	adantr 481	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ((𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅) ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅)))
21	15, 20	mpbird 247	. . . . . . 7 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅))
22		fveq2 6191	. . . . . . . . 9 ⊢ (𝑔 = 𝑝 → (iEdg‘𝑔) = (iEdg‘𝑝))
23	22	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑔 = 𝑝 → ((iEdg‘𝑔) = ∅ ↔ (iEdg‘𝑝) = ∅))
24	23	elrab 3363	. . . . . . 7 ⊢ (𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ↔ (𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅))
25	21, 24	sylibr 224	. . . . . 6 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
26	25	exlimivv 1860	. . . . 5 ⊢ (∃𝑣∃𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
27	1, 26	sylbi 207	. . . 4 ⊢ (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
28	27	ssriv 3607	. . 3 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
29		eqid 2622	. . . 4 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
30	29	griedg0prc 26156	. . 3 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V
31		prcssprc 4806	. . 3 ⊢ (({⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∧ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V) → {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V)
32	28, 30, 31	mp2an 708	. 2 ⊢ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V
33		df-3an 1039	. . . . . . . 8 ⊢ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0) ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0))
34	33	bicomi 214	. . . . . . 7 ⊢ (((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0) ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0))
35	34	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)))
36		0xnn0 11369	. . . . . . 7 ⊢ 0 ∈ ℕ0*
37		ibar 525	. . . . . . 7 ⊢ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
38	36, 37	mpan2 707	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
39		eqid 2622	. . . . . . . 8 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
40		eqid 2622	. . . . . . . 8 ⊢ (VtxDeg‘𝑔) = (VtxDeg‘𝑔)
41	39, 40	isrusgr0 26462	. . . . . . 7 ⊢ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) → (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
42	36, 41	mpan2 707	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
43	35, 38, 42	3bitr4d 300	. . . . 5 ⊢ (𝑔 ∈ USGraph → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ 𝑔 RegUSGraph 0))
44	43	rabbiia 3185	. . . 4 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0}
45		usgr0edg0rusgr 26471	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (Edg‘𝑔) = ∅))
46		usgruhgr 26078	. . . . . . 7 ⊢ (𝑔 ∈ USGraph → 𝑔 ∈ UHGraph )
47		uhgriedg0edg0 26022	. . . . . . 7 ⊢ (𝑔 ∈ UHGraph → ((Edg‘𝑔) = ∅ ↔ (iEdg‘𝑔) = ∅))
48	46, 47	syl 17	. . . . . 6 ⊢ (𝑔 ∈ USGraph → ((Edg‘𝑔) = ∅ ↔ (iEdg‘𝑔) = ∅))
49	45, 48	bitrd 268	. . . . 5 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (iEdg‘𝑔) = ∅))
50	49	rabbiia 3185	. . . 4 ⊢ {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
51	44, 50	eqtri 2644	. . 3 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
52		neleq1 2902	. . 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))
53	51, 52	ax-mp 5	. 2 ⊢ ({𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V ↔ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V)
54	32, 53	mpbir 221	1 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V

Syntax hints:   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ∉ wnel 2897  ∀wral 2912  {crab 2916  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ⟨cop 4183   class class class wbr 4653  {copab 4712  ⟶wf 5884  ‘cfv 5888  0cc0 9936  ℕ₀^*cxnn0 11363  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   USGraph cusgr 26044  VtxDegcvtxdg 26361   RegUSGraph crusgr 26452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-xadd 11947  df-fz 12327  df-hash 13118  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-usgr 26046  df-vtxdg 26362  df-rgr 26453  df-rusgr 26454

This theorem is referenced by:  rusgrprc  26486