Theorem rgrusgrprc 29683

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 5476	. . . . 5 ⊢ (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣∃𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
2		f0bi 6717	. . . . . . . . . 10 ⊢ (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
3		opeq2 4812	. . . . . . . . . . . 12 ⊢ (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ = ⟨𝑣, ∅⟩)
4		usgr0eop 29340	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ V → ⟨𝑣, ∅⟩ ∈ USGraph)
5	4	elv 3437	. . . . . . . . . . . 12 ⊢ ⟨𝑣, ∅⟩ ∈ USGraph
6	3, 5	eqeltrdi 2848	. . . . . . . . . . 11 ⊢ (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph)
7		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
8		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑒 ∈ V
9	7, 8	opiedgfvi 29104	. . . . . . . . . . . 12 ⊢ (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
10		id 22	. . . . . . . . . . . 12 ⊢ (𝑒 = ∅ → 𝑒 = ∅)
11	9, 10	eqtrid 2787	. . . . . . . . . . 11 ⊢ (𝑒 = ∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
12	6, 11	jca 516	. . . . . . . . . 10 ⊢ (𝑒 = ∅ → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
13	2, 12	sylbi 218	. . . . . . . . 9 ⊢ (𝑒:∅⟶∅ → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
14	13	adantl 482	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
15		eleq1 2828	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → (𝑝 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph))
16		fveqeq2 6843	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → ((iEdg‘𝑝) = ∅ ↔ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
17	15, 16	anbi12d 638	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑣, 𝑒⟩ → ((𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅) ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅)))
18	17	adantr 481	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ((𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅) ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅)))
19	14, 18	mpbird 258	. . . . . . 7 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅))
20		fveqeq2 6843	. . . . . . . 8 ⊢ (𝑔 = 𝑝 → ((iEdg‘𝑔) = ∅ ↔ (iEdg‘𝑝) = ∅))
21	20	elrab 3636	. . . . . . 7 ⊢ (𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ↔ (𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅))
22	19, 21	sylibr 235	. . . . . 6 ⊢ ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
23	22	exlimivv 1939	. . . . 5 ⊢ (∃𝑣∃𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
24	1, 23	sylbi 218	. . . 4 ⊢ (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
25	24	ssriv 3926	. . 3 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
26		eqid 2740	. . . 4 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
27	26	griedg0prc 29358	. . 3 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V
28		prcssprc 5262	. . 3 ⊢ (({⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∧ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V) → {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V)
29	25, 27, 28	mp2an 698	. 2 ⊢ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V
30		df-3an 1094	. . . . . . . 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 225	. . . . . . 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 12514	. . . . . . 7 ⊢ 0 ∈ ℕ0*
34		ibar 533	. . . . . . 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 697	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
36		eqid 2740	. . . . . . . 8 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
37		eqid 2740	. . . . . . . 8 ⊢ (VtxDeg‘𝑔) = (VtxDeg‘𝑔)
38	36, 37	isrusgr0 29660	. . . . . . 7 ⊢ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) → (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
39	33, 38	mpan2 697	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
40	32, 35, 39	3bitr4d 312	. . . . 5 ⊢ (𝑔 ∈ USGraph → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ 𝑔 RegUSGraph 0))
41	40	rabbiia 3396	. . . 4 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0}
42		usgr0edg0rusgr 29669	. . . . . 6 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (Edg‘𝑔) = ∅))
43		usgruhgr 29280	. . . . . . 7 ⊢ (𝑔 ∈ USGraph → 𝑔 ∈ UHGraph)
44		uhgriedg0edg0 29221	. . . . . . 7 ⊢ (𝑔 ∈ UHGraph → ((Edg‘𝑔) = ∅ ↔ (iEdg‘𝑔) = ∅))
45	43, 44	syl 17	. . . . . 6 ⊢ (𝑔 ∈ USGraph → ((Edg‘𝑔) = ∅ ↔ (iEdg‘𝑔) = ∅))
46	42, 45	bitrd 280	. . . . 5 ⊢ (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (iEdg‘𝑔) = ∅))
47	46	rabbiia 3396	. . . 4 ⊢ {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
48	41, 47	eqtri 2763	. . 3 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
49		neleq1 3045	. . 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 232	1 ⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ∉ wnel 3039  ∀wral 3054  {crab 3392  Vcvv 3432   ⊆ wss 3890  ∅c0 4268  ⟨cop 4568   class class class wbr 5079  {copab 5141  ⟶wf 6488  ‘cfv 6492  0cc0 11036  ℕ₀^*cxnn0 12508  Vtxcvtx 29090  iEdgciedg 29091  Edgcedg 29141  UHGraphcuhgr 29150  USGraphcusgr 29243  VtxDegcvtxdg 29559   RegUSGraph crusgr 29650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-n0 12436  df-xnn0 12509  df-z 12523  df-uz 12787  df-xadd 13062  df-fz 13460  df-hash 14291  df-iedg 29093  df-edg 29142  df-uhgr 29152  df-upgr 29176  df-uspgr 29244  df-usgr 29245  df-vtxdg 29560  df-rgr 29651  df-rusgr 29652

This theorem is referenced by:  rusgrprc  29684