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Theorem rgrusgrprc 26712
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 5178 . . . . 5 (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
2 f0bi 6299 . . . . . . . . . 10 (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
3 opeq2 4596 . . . . . . . . . . . 12 (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ = ⟨𝑣, ∅⟩)
4 vex 3394 . . . . . . . . . . . . 13 𝑣 ∈ V
5 usgr0eop 26353 . . . . . . . . . . . . 13 (𝑣 ∈ V → ⟨𝑣, ∅⟩ ∈ USGraph)
64, 5ax-mp 5 . . . . . . . . . . . 12 𝑣, ∅⟩ ∈ USGraph
73, 6syl6eqel 2893 . . . . . . . . . . 11 (𝑒 = ∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph)
8 vex 3394 . . . . . . . . . . . . 13 𝑒 ∈ V
9 opiedgfv 26100 . . . . . . . . . . . . 13 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒)
104, 8, 9mp2an 675 . . . . . . . . . . . 12 (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
11 id 22 . . . . . . . . . . . 12 (𝑒 = ∅ → 𝑒 = ∅)
1210, 11syl5eq 2852 . . . . . . . . . . 11 (𝑒 = ∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
137, 12jca 503 . . . . . . . . . 10 (𝑒 = ∅ → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
142, 13sylbi 208 . . . . . . . . 9 (𝑒:∅⟶∅ → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
1514adantl 469 . . . . . . . 8 ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
16 eleq1 2873 . . . . . . . . . 10 (𝑝 = ⟨𝑣, 𝑒⟩ → (𝑝 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph))
17 fveqeq2 6413 . . . . . . . . . 10 (𝑝 = ⟨𝑣, 𝑒⟩ → ((iEdg‘𝑝) = ∅ ↔ (iEdg‘⟨𝑣, 𝑒⟩) = ∅))
1816, 17anbi12d 618 . . . . . . . . 9 (𝑝 = ⟨𝑣, 𝑒⟩ → ((𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅) ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅)))
1918adantr 468 . . . . . . . 8 ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ((𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅) ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (iEdg‘⟨𝑣, 𝑒⟩) = ∅)))
2015, 19mpbird 248 . . . . . . 7 ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅))
21 fveqeq2 6413 . . . . . . . 8 (𝑔 = 𝑝 → ((iEdg‘𝑔) = ∅ ↔ (iEdg‘𝑝) = ∅))
2221elrab 3559 . . . . . . 7 (𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ↔ (𝑝 ∈ USGraph ∧ (iEdg‘𝑝) = ∅))
2320, 22sylibr 225 . . . . . 6 ((𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
2423exlimivv 2023 . . . . 5 (∃𝑣𝑒(𝑝 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
251, 24sylbi 208 . . . 4 (𝑝 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} → 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅})
2625ssriv 3802 . . 3 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
27 eqid 2806 . . . 4 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
2827griedg0prc 26371 . . 3 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V
29 prcssprc 5001 . . 3 (({⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∧ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V) → {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V)
3026, 28, 29mp2an 675 . 2 {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V
31 df-3an 1102 . . . . . . . 8 ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0) ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0))
3231bicomi 215 . . . . . . 7 (((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0) ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0))
3332a1i 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)))
34 0xnn0 11631 . . . . . . 7 0 ∈ ℕ0*
35 ibar 520 . . . . . . 7 ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
3634, 35mpan2 674 . . . . . 6 (𝑔 ∈ USGraph → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
37 eqid 2806 . . . . . . . 8 (Vtx‘𝑔) = (Vtx‘𝑔)
38 eqid 2806 . . . . . . . 8 (VtxDeg‘𝑔) = (VtxDeg‘𝑔)
3937, 38isrusgr0 26689 . . . . . . 7 ((𝑔 ∈ USGraph ∧ 0 ∈ ℕ0*) → (𝑔RegUSGraph0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
4034, 39mpan2 674 . . . . . 6 (𝑔 ∈ USGraph → (𝑔RegUSGraph0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0)))
4133, 36, 403bitr4d 302 . . . . 5 (𝑔 ∈ USGraph → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0 ↔ 𝑔RegUSGraph0))
4241rabbiia 3374 . . . 4 {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ 𝑔RegUSGraph0}
43 usgr0edg0rusgr 26698 . . . . . 6 (𝑔 ∈ USGraph → (𝑔RegUSGraph0 ↔ (Edg‘𝑔) = ∅))
44 usgruhgr 26292 . . . . . . 7 (𝑔 ∈ USGraph → 𝑔 ∈ UHGraph)
45 uhgriedg0edg0 26235 . . . . . . 7 (𝑔 ∈ UHGraph → ((Edg‘𝑔) = ∅ ↔ (iEdg‘𝑔) = ∅))
4644, 45syl 17 . . . . . 6 (𝑔 ∈ USGraph → ((Edg‘𝑔) = ∅ ↔ (iEdg‘𝑔) = ∅))
4743, 46bitrd 270 . . . . 5 (𝑔 ∈ USGraph → (𝑔RegUSGraph0 ↔ (iEdg‘𝑔) = ∅))
4847rabbiia 3374 . . . 4 {𝑔 ∈ USGraph ∣ 𝑔RegUSGraph0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
4942, 48eqtri 2828 . . 3 {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} = {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅}
50 neleq1 3086 . . 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))
5149, 50ax-mp 5 . 2 ({𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V ↔ {𝑔 ∈ USGraph ∣ (iEdg‘𝑔) = ∅} ∉ V)
5230, 51mpbir 222 1 {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2156  wnel 3081  wral 3096  {crab 3100  Vcvv 3391  wss 3769  c0 4116  cop 4376   class class class wbr 4844  {copab 4906  wf 6093  cfv 6097  0cc0 10217  0*cxnn0 11625  Vtxcvtx 26087  iEdgciedg 26088  Edgcedg 26152  UHGraphcuhgr 26164  USGraphcusgr 26258  VtxDegcvtxdg 26588  RegUSGraphcrusgr 26679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-cnex 10273  ax-resscn 10274  ax-1cn 10275  ax-icn 10276  ax-addcl 10277  ax-addrcl 10278  ax-mulcl 10279  ax-mulrcl 10280  ax-mulcom 10281  ax-addass 10282  ax-mulass 10283  ax-distr 10284  ax-i2m1 10285  ax-1ne0 10286  ax-1rid 10287  ax-rnegex 10288  ax-rrecex 10289  ax-cnre 10290  ax-pre-lttri 10291  ax-pre-lttrn 10292  ax-pre-ltadd 10293  ax-pre-mulgt0 10294
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-1st 7394  df-2nd 7395  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-1o 7792  df-er 7975  df-en 8189  df-dom 8190  df-sdom 8191  df-fin 8192  df-card 9044  df-pnf 10357  df-mnf 10358  df-xr 10359  df-ltxr 10360  df-le 10361  df-sub 10549  df-neg 10550  df-nn 11302  df-2 11360  df-n0 11556  df-xnn0 11626  df-z 11640  df-uz 11901  df-xadd 12159  df-fz 12546  df-hash 13334  df-iedg 26090  df-edg 26153  df-uhgr 26166  df-upgr 26190  df-uspgr 26259  df-usgr 26260  df-vtxdg 26589  df-rgr 26680  df-rusgr 26681
This theorem is referenced by:  rusgrprc  26713
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