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Theorem rgrusgrprc 28846
Description: The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
Assertion
Ref Expression
rgrusgrprc {𝑔 ∈ USGraph ∣ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0} βˆ‰ 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 5528 . . . . 5 (𝑝 ∈ {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} ↔ βˆƒπ‘£βˆƒπ‘’(𝑝 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…))
2 f0bi 6775 . . . . . . . . . 10 (𝑒:βˆ…βŸΆβˆ… ↔ 𝑒 = βˆ…)
3 opeq2 4875 . . . . . . . . . . . 12 (𝑒 = βˆ… β†’ βŸ¨π‘£, π‘’βŸ© = βŸ¨π‘£, βˆ…βŸ©)
4 usgr0eop 28503 . . . . . . . . . . . . 13 (𝑣 ∈ V β†’ βŸ¨π‘£, βˆ…βŸ© ∈ USGraph)
54elv 3481 . . . . . . . . . . . 12 βŸ¨π‘£, βˆ…βŸ© ∈ USGraph
63, 5eqeltrdi 2842 . . . . . . . . . . 11 (𝑒 = βˆ… β†’ βŸ¨π‘£, π‘’βŸ© ∈ USGraph)
7 vex 3479 . . . . . . . . . . . . 13 𝑣 ∈ V
8 vex 3479 . . . . . . . . . . . . 13 𝑒 ∈ V
97, 8opiedgfvi 28270 . . . . . . . . . . . 12 (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = 𝑒
10 id 22 . . . . . . . . . . . 12 (𝑒 = βˆ… β†’ 𝑒 = βˆ…)
119, 10eqtrid 2785 . . . . . . . . . . 11 (𝑒 = βˆ… β†’ (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = βˆ…)
126, 11jca 513 . . . . . . . . . 10 (𝑒 = βˆ… β†’ (βŸ¨π‘£, π‘’βŸ© ∈ USGraph ∧ (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = βˆ…))
132, 12sylbi 216 . . . . . . . . 9 (𝑒:βˆ…βŸΆβˆ… β†’ (βŸ¨π‘£, π‘’βŸ© ∈ USGraph ∧ (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = βˆ…))
1413adantl 483 . . . . . . . 8 ((𝑝 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…) β†’ (βŸ¨π‘£, π‘’βŸ© ∈ USGraph ∧ (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = βˆ…))
15 eleq1 2822 . . . . . . . . . 10 (𝑝 = βŸ¨π‘£, π‘’βŸ© β†’ (𝑝 ∈ USGraph ↔ βŸ¨π‘£, π‘’βŸ© ∈ USGraph))
16 fveqeq2 6901 . . . . . . . . . 10 (𝑝 = βŸ¨π‘£, π‘’βŸ© β†’ ((iEdgβ€˜π‘) = βˆ… ↔ (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = βˆ…))
1715, 16anbi12d 632 . . . . . . . . 9 (𝑝 = βŸ¨π‘£, π‘’βŸ© β†’ ((𝑝 ∈ USGraph ∧ (iEdgβ€˜π‘) = βˆ…) ↔ (βŸ¨π‘£, π‘’βŸ© ∈ USGraph ∧ (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = βˆ…)))
1817adantr 482 . . . . . . . 8 ((𝑝 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…) β†’ ((𝑝 ∈ USGraph ∧ (iEdgβ€˜π‘) = βˆ…) ↔ (βŸ¨π‘£, π‘’βŸ© ∈ USGraph ∧ (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = βˆ…)))
1914, 18mpbird 257 . . . . . . 7 ((𝑝 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…) β†’ (𝑝 ∈ USGraph ∧ (iEdgβ€˜π‘) = βˆ…))
20 fveqeq2 6901 . . . . . . . 8 (𝑔 = 𝑝 β†’ ((iEdgβ€˜π‘”) = βˆ… ↔ (iEdgβ€˜π‘) = βˆ…))
2120elrab 3684 . . . . . . 7 (𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…} ↔ (𝑝 ∈ USGraph ∧ (iEdgβ€˜π‘) = βˆ…))
2219, 21sylibr 233 . . . . . 6 ((𝑝 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…) β†’ 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…})
2322exlimivv 1936 . . . . 5 (βˆƒπ‘£βˆƒπ‘’(𝑝 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…) β†’ 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…})
241, 23sylbi 216 . . . 4 (𝑝 ∈ {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} β†’ 𝑝 ∈ {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…})
2524ssriv 3987 . . 3 {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βŠ† {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…}
26 eqid 2733 . . . 4 {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} = {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…}
2726griedg0prc 28521 . . 3 {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βˆ‰ V
28 prcssprc 5326 . . 3 (({βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βŠ† {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…} ∧ {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βˆ‰ V) β†’ {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…} βˆ‰ V)
2925, 27, 28mp2an 691 . 2 {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…} βˆ‰ V
30 df-3an 1090 . . . . . . . 8 ((𝑔 ∈ USGraph ∧ 0 ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0) ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ β„•0*) ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0))
3130bicomi 223 . . . . . . 7 (((𝑔 ∈ USGraph ∧ 0 ∈ β„•0*) ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0) ↔ (𝑔 ∈ USGraph ∧ 0 ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0))
3231a1i 11 . . . . . 6 (𝑔 ∈ USGraph β†’ (((𝑔 ∈ USGraph ∧ 0 ∈ β„•0*) ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0) ↔ (𝑔 ∈ USGraph ∧ 0 ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0)))
33 0xnn0 12550 . . . . . . 7 0 ∈ β„•0*
34 ibar 530 . . . . . . 7 ((𝑔 ∈ USGraph ∧ 0 ∈ β„•0*) β†’ (βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0 ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ β„•0*) ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0)))
3533, 34mpan2 690 . . . . . 6 (𝑔 ∈ USGraph β†’ (βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0 ↔ ((𝑔 ∈ USGraph ∧ 0 ∈ β„•0*) ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0)))
36 eqid 2733 . . . . . . . 8 (Vtxβ€˜π‘”) = (Vtxβ€˜π‘”)
37 eqid 2733 . . . . . . . 8 (VtxDegβ€˜π‘”) = (VtxDegβ€˜π‘”)
3836, 37isrusgr0 28823 . . . . . . 7 ((𝑔 ∈ USGraph ∧ 0 ∈ β„•0*) β†’ (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0)))
3933, 38mpan2 690 . . . . . 6 (𝑔 ∈ USGraph β†’ (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0)))
4032, 35, 393bitr4d 311 . . . . 5 (𝑔 ∈ USGraph β†’ (βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0 ↔ 𝑔 RegUSGraph 0))
4140rabbiia 3437 . . . 4 {𝑔 ∈ USGraph ∣ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0} = {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0}
42 usgr0edg0rusgr 28832 . . . . . 6 (𝑔 ∈ USGraph β†’ (𝑔 RegUSGraph 0 ↔ (Edgβ€˜π‘”) = βˆ…))
43 usgruhgr 28443 . . . . . . 7 (𝑔 ∈ USGraph β†’ 𝑔 ∈ UHGraph)
44 uhgriedg0edg0 28387 . . . . . . 7 (𝑔 ∈ UHGraph β†’ ((Edgβ€˜π‘”) = βˆ… ↔ (iEdgβ€˜π‘”) = βˆ…))
4543, 44syl 17 . . . . . 6 (𝑔 ∈ USGraph β†’ ((Edgβ€˜π‘”) = βˆ… ↔ (iEdgβ€˜π‘”) = βˆ…))
4642, 45bitrd 279 . . . . 5 (𝑔 ∈ USGraph β†’ (𝑔 RegUSGraph 0 ↔ (iEdgβ€˜π‘”) = βˆ…))
4746rabbiia 3437 . . . 4 {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0} = {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…}
4841, 47eqtri 2761 . . 3 {𝑔 ∈ USGraph ∣ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0} = {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…}
49 neleq1 3053 . . 3 ({𝑔 ∈ USGraph ∣ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0} = {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…} β†’ ({𝑔 ∈ USGraph ∣ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0} βˆ‰ V ↔ {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…} βˆ‰ V))
5048, 49ax-mp 5 . 2 ({𝑔 ∈ USGraph ∣ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0} βˆ‰ V ↔ {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…} βˆ‰ V)
5129, 50mpbir 230 1 {𝑔 ∈ USGraph ∣ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0} βˆ‰ V
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   βˆ‰ wnel 3047  βˆ€wral 3062  {crab 3433  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323  βŸ¨cop 4635   class class class wbr 5149  {copab 5211  βŸΆwf 6540  β€˜cfv 6544  0cc0 11110  β„•0*cxnn0 12544  Vtxcvtx 28256  iEdgciedg 28257  Edgcedg 28307  UHGraphcuhgr 28316  USGraphcusgr 28409  VtxDegcvtxdg 28722   RegUSGraph crusgr 28813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-xadd 13093  df-fz 13485  df-hash 14291  df-iedg 28259  df-edg 28308  df-uhgr 28318  df-upgr 28342  df-uspgr 28410  df-usgr 28411  df-vtxdg 28723  df-rgr 28814  df-rusgr 28815
This theorem is referenced by:  rusgrprc  28847
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