MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rgrusgrprc Structured version   Visualization version   GIF version

Theorem rgrusgrprc 28884
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 5527 . . . . 5 (𝑝 ∈ {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} ↔ βˆƒπ‘£βˆƒπ‘’(𝑝 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…))
2 f0bi 6774 . . . . . . . . . 10 (𝑒:βˆ…βŸΆβˆ… ↔ 𝑒 = βˆ…)
3 opeq2 4874 . . . . . . . . . . . 12 (𝑒 = βˆ… β†’ βŸ¨π‘£, π‘’βŸ© = βŸ¨π‘£, βˆ…βŸ©)
4 usgr0eop 28541 . . . . . . . . . . . . 13 (𝑣 ∈ V β†’ βŸ¨π‘£, βˆ…βŸ© ∈ USGraph)
54elv 3480 . . . . . . . . . . . 12 βŸ¨π‘£, βˆ…βŸ© ∈ USGraph
63, 5eqeltrdi 2841 . . . . . . . . . . 11 (𝑒 = βˆ… β†’ βŸ¨π‘£, π‘’βŸ© ∈ USGraph)
7 vex 3478 . . . . . . . . . . . . 13 𝑣 ∈ V
8 vex 3478 . . . . . . . . . . . . 13 𝑒 ∈ V
97, 8opiedgfvi 28308 . . . . . . . . . . . 12 (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = 𝑒
10 id 22 . . . . . . . . . . . 12 (𝑒 = βˆ… β†’ 𝑒 = βˆ…)
119, 10eqtrid 2784 . . . . . . . . . . 11 (𝑒 = βˆ… β†’ (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = βˆ…)
126, 11jca 512 . . . . . . . . . 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 482 . . . . . . . 8 ((𝑝 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…) β†’ (βŸ¨π‘£, π‘’βŸ© ∈ USGraph ∧ (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = βˆ…))
15 eleq1 2821 . . . . . . . . . 10 (𝑝 = βŸ¨π‘£, π‘’βŸ© β†’ (𝑝 ∈ USGraph ↔ βŸ¨π‘£, π‘’βŸ© ∈ USGraph))
16 fveqeq2 6900 . . . . . . . . . 10 (𝑝 = βŸ¨π‘£, π‘’βŸ© β†’ ((iEdgβ€˜π‘) = βˆ… ↔ (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = βˆ…))
1715, 16anbi12d 631 . . . . . . . . 9 (𝑝 = βŸ¨π‘£, π‘’βŸ© β†’ ((𝑝 ∈ USGraph ∧ (iEdgβ€˜π‘) = βˆ…) ↔ (βŸ¨π‘£, π‘’βŸ© ∈ USGraph ∧ (iEdgβ€˜βŸ¨π‘£, π‘’βŸ©) = βˆ…)))
1817adantr 481 . . . . . . . 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 256 . . . . . . 7 ((𝑝 = βŸ¨π‘£, π‘’βŸ© ∧ 𝑒:βˆ…βŸΆβˆ…) β†’ (𝑝 ∈ USGraph ∧ (iEdgβ€˜π‘) = βˆ…))
20 fveqeq2 6900 . . . . . . . 8 (𝑔 = 𝑝 β†’ ((iEdgβ€˜π‘”) = βˆ… ↔ (iEdgβ€˜π‘) = βˆ…))
2120elrab 3683 . . . . . . 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 1935 . . . . 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 3986 . . 3 {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βŠ† {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…}
26 eqid 2732 . . . 4 {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} = {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…}
2726griedg0prc 28559 . . 3 {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βˆ‰ V
28 prcssprc 5325 . . 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 690 . 2 {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…} βˆ‰ V
30 df-3an 1089 . . . . . . . 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 12552 . . . . . . 7 0 ∈ β„•0*
34 ibar 529 . . . . . . 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 689 . . . . . 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 2732 . . . . . . . 8 (Vtxβ€˜π‘”) = (Vtxβ€˜π‘”)
37 eqid 2732 . . . . . . . 8 (VtxDegβ€˜π‘”) = (VtxDegβ€˜π‘”)
3836, 37isrusgr0 28861 . . . . . . 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 689 . . . . . 6 (𝑔 ∈ USGraph β†’ (𝑔 RegUSGraph 0 ↔ (𝑔 ∈ USGraph ∧ 0 ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0)))
4032, 35, 393bitr4d 310 . . . . 5 (𝑔 ∈ USGraph β†’ (βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0 ↔ 𝑔 RegUSGraph 0))
4140rabbiia 3436 . . . 4 {𝑔 ∈ USGraph ∣ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0} = {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0}
42 usgr0edg0rusgr 28870 . . . . . 6 (𝑔 ∈ USGraph β†’ (𝑔 RegUSGraph 0 ↔ (Edgβ€˜π‘”) = βˆ…))
43 usgruhgr 28481 . . . . . . 7 (𝑔 ∈ USGraph β†’ 𝑔 ∈ UHGraph)
44 uhgriedg0edg0 28425 . . . . . . 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 278 . . . . 5 (𝑔 ∈ USGraph β†’ (𝑔 RegUSGraph 0 ↔ (iEdgβ€˜π‘”) = βˆ…))
4746rabbiia 3436 . . . 4 {𝑔 ∈ USGraph ∣ 𝑔 RegUSGraph 0} = {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…}
4841, 47eqtri 2760 . . 3 {𝑔 ∈ USGraph ∣ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = 0} = {𝑔 ∈ USGraph ∣ (iEdgβ€˜π‘”) = βˆ…}
49 neleq1 3052 . . 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 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   βˆ‰ wnel 3046  βˆ€wral 3061  {crab 3432  Vcvv 3474   βŠ† wss 3948  βˆ…c0 4322  βŸ¨cop 4634   class class class wbr 5148  {copab 5210  βŸΆwf 6539  β€˜cfv 6543  0cc0 11112  β„•0*cxnn0 12546  Vtxcvtx 28294  iEdgciedg 28295  Edgcedg 28345  UHGraphcuhgr 28354  USGraphcusgr 28447  VtxDegcvtxdg 28760   RegUSGraph crusgr 28851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-n0 12475  df-xnn0 12547  df-z 12561  df-uz 12825  df-xadd 13095  df-fz 13487  df-hash 14293  df-iedg 28297  df-edg 28346  df-uhgr 28356  df-upgr 28380  df-uspgr 28448  df-usgr 28449  df-vtxdg 28761  df-rgr 28852  df-rusgr 28853
This theorem is referenced by:  rusgrprc  28885
  Copyright terms: Public domain W3C validator