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Theorem rgrusgrprc 26355
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.
StepHypRef Expression
1 elopab 4943 . . . . 5 (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
2 f0bi 6045 . . . . . . . . . 10 (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
3 opeq2 4371 . . . . . . . . . . . 12 (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ = ⟨𝑣, ∅⟩)
4 vex 3189 . . . . . . . . . . . . 13 𝑣 ∈ V
5 usgr0eop 26031 . . . . . . . . . . . . 13 (𝑣 ∈ V → ⟨𝑣, ∅⟩ ∈ USGraph )
64, 5ax-mp 5 . . . . . . . . . . . 12 𝑣, ∅⟩ ∈ USGraph
73, 6syl6eqel 2706 . . . . . . . . . . 11 (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph )
8 vex 3189 . . . . . . . . . . . . 13 𝑒 ∈ V
9 opiedgfv 25787 . . . . . . . . . . . . 13 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒)
104, 8, 9mp2an 707 . . . . . . . . . . . 12 (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
11 id 22 . . . . . . . . . . . 12 (𝑒 = ∅ → 𝑒 = ∅)
1210, 11syl5eq 2667 . . . . . . . . . . 11 (𝑒 = ∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
137, 12jca 554 . . . . . . . . . 10 (𝑒 = ∅ → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
142, 13sylbi 207 . . . . . . . . 9 (𝑒:∅⟶∅ → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
1514adantl 482 . . . . . . . 8 ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
16 eleq1 2686 . . . . . . . . . 10 (𝑝 = ⟨𝑣, 𝑒⟩ → (𝑝 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph ))
17 fveq2 6148 . . . . . . . . . . 11 (𝑝 = ⟨𝑣, 𝑒⟩ → (iEdg‘𝑝) = (iEdg‘⟨𝑣, 𝑒⟩))
1817eqeq1d 2623 . . . . . . . . . 10 (𝑝 = ⟨𝑣, 𝑒⟩ → ((iEdg‘𝑝) = ∅ ↔ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
1916, 18anbi12d 746 . . . . . . . . 9 (𝑝 = ⟨𝑣, 𝑒⟩ → ((𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅) ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅)))
2019adantr 481 . . . . . . . 8 ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ((𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅) ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅)))
2115, 20mpbird 247 . . . . . . 7 ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅))
22 fveq2 6148 . . . . . . . . 9 (𝑔 = 𝑝 → (iEdg‘𝑔) = (iEdg‘𝑝))
2322eqeq1d 2623 . . . . . . . 8 (𝑔 = 𝑝 → ((iEdg‘𝑔) = ∅ ↔ (iEdg‘𝑝) = ∅))
2423elrab 3346 . . . . . . 7 (𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ↔ (𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅))
2521, 24sylibr 224 . . . . . 6 ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
2625exlimivv 1857 . . . . 5 (∃𝑣𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
271, 26sylbi 207 . . . 4 (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
2827ssriv 3587 . . 3 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
29 eqid 2621 . . . 4 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
3029griedg0prc 26049 . . 3 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V
31 prcssprc 4766 . . 3 (({⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∧ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V) → {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V)
3228, 30, 31mp2an 707 . 2 {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V
33 df-3an 1038 . . . . . . . 8 ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0) ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0))
3433bicomi 214 . . . . . . 7 (((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0) ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0))
3534a1i 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 11313 . . . . . . 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)))
3836, 37mpan2 706 . . . . . 6 (𝑔 ∈ USGraph → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
39 eqid 2621 . . . . . . . 8 (Vtx‘𝑔) = (Vtx‘𝑔)
40 eqid 2621 . . . . . . . 8 (VtxDeg‘𝑔) = (VtxDeg‘𝑔)
4139, 40isrusgr0 26332 . . . . . . 7 ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) → (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
4236, 41mpan2 706 . . . . . 6 (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
4335, 38, 423bitr4d 300 . . . . 5 (𝑔 ∈ USGraph → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ 𝑔 RegUSGraph 0))
4443rabbiia 3173 . . . 4 {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0}
45 usgr0edg0rusgr 26341 . . . . . 6 (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (Edg‘𝑔) = ∅))
46 usgruhgr 25971 . . . . . . 7 (𝑔 ∈ USGraph → 𝑔 ∈ UHGraph )
47 uhgriedg0edg0 25917 . . . . . . 7 (𝑔 ∈ UHGraph → ((Edg‘𝑔) = ∅ ↔ (iEdg‘𝑔) = ∅))
4846, 47syl 17 . . . . . 6 (𝑔 ∈ USGraph → ((Edg‘𝑔) = ∅ ↔ (iEdg‘𝑔) = ∅))
4945, 48bitrd 268 . . . . 5 (𝑔 ∈ USGraph → (𝑔 RegUSGraph 0 ↔ (iEdg‘𝑔) = ∅))
5049rabbiia 3173 . . . 4 {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
5144, 50eqtri 2643 . . 3 {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
52 neleq1 2898 . . 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))
5351, 52ax-mp 5 . 2 ({𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V ↔ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V)
5432, 53mpbir 221 1 {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wnel 2893  wral 2907  {crab 2911  Vcvv 3186  wss 3555  c0 3891  cop 4154   class class class wbr 4613  {copab 4672  wf 5843  cfv 5847  0cc0 9880  0*cxnn0 11307  Vtxcvtx 25774  iEdgciedg 25775  Edgcedg 25839   UHGraph cuhgr 25847   USGraph cusgr 25937  VtxDegcvtxdg 26248   RegUSGraph crusgr 26322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-xadd 11891  df-fz 12269  df-hash 13058  df-iedg 25777  df-edg 25840  df-uhgr 25849  df-upgr 25873  df-uspgr 25938  df-usgr 25939  df-vtxdg 26249  df-rgr 26323  df-rusgr 26324
This theorem is referenced by:  rusgrprc  26356
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