Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nbuhgr2vtx1edgb Structured version   Visualization version   Unicode version

Theorem nbuhgr2vtx1edgb 39584
 Description: If a hypergraph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.)
Hypotheses
Ref Expression
nbgr2vtx1edg.v Vtx
nbgr2vtx1edg.e Edg
Assertion
Ref Expression
nbuhgr2vtx1edgb UHGraph NeighbVtx
Distinct variable groups:   ,   ,,   ,,
Allowed substitution hint:   ()

Proof of Theorem nbuhgr2vtx1edgb
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nbgr2vtx1edg.v . . . . 5 Vtx
2 fvex 5889 . . . . 5 Vtx
31, 2eqeltri 2545 . . . 4
4 hash2prb 12674 . . . 4
53, 4ax-mp 5 . . 3
6 simpr 468 . . . . . . . . . . . 12 UHGraph
76ancomd 458 . . . . . . . . . . 11 UHGraph
87ad2antrr 740 . . . . . . . . . 10 UHGraph
9 id 22 . . . . . . . . . . . . 13
109necomd 2698 . . . . . . . . . . . 12
1110adantr 472 . . . . . . . . . . 11
1211ad2antlr 741 . . . . . . . . . 10 UHGraph
13 prcom 4041 . . . . . . . . . . . . . 14
1413eleq1i 2540 . . . . . . . . . . . . 13
1514biimpi 199 . . . . . . . . . . . 12
16 sseq2 3440 . . . . . . . . . . . . 13
1716adantl 473 . . . . . . . . . . . 12
1813eqimssi 3472 . . . . . . . . . . . . 13
1918a1i 11 . . . . . . . . . . . 12
2015, 17, 19rspcedvd 3143 . . . . . . . . . . 11
2120adantl 473 . . . . . . . . . 10 UHGraph
22 nbgr2vtx1edg.e . . . . . . . . . . . 12 Edg
231, 22nbgrel 39574 . . . . . . . . . . 11 UHGraph NeighbVtx
2423ad3antrrr 744 . . . . . . . . . 10 UHGraph NeighbVtx
258, 12, 21, 24mpbir3and 1213 . . . . . . . . 9 UHGraph NeighbVtx
266ad2antrr 740 . . . . . . . . . 10 UHGraph
27 simplrl 778 . . . . . . . . . 10 UHGraph
28 id 22 . . . . . . . . . . . 12
29 sseq2 3440 . . . . . . . . . . . . 13
3029adantl 473 . . . . . . . . . . . 12
31 prcom 4041 . . . . . . . . . . . . . 14
3231eqimssi 3472 . . . . . . . . . . . . 13
3332a1i 11 . . . . . . . . . . . 12
3428, 30, 33rspcedvd 3143 . . . . . . . . . . 11
3534adantl 473 . . . . . . . . . 10 UHGraph
361, 22nbgrel 39574 . . . . . . . . . . 11 UHGraph NeighbVtx
3736ad3antrrr 744 . . . . . . . . . 10 UHGraph NeighbVtx
3826, 27, 35, 37mpbir3and 1213 . . . . . . . . 9 UHGraph NeighbVtx
3925, 38jca 541 . . . . . . . 8 UHGraph NeighbVtx NeighbVtx
4039ex 441 . . . . . . 7 UHGraph NeighbVtx NeighbVtx
411, 22nbuhgr2vtx1edgblem 39583 . . . . . . . . . . . 12 UHGraph NeighbVtx
42413exp 1230 . . . . . . . . . . 11 UHGraph NeighbVtx
4342adantr 472 . . . . . . . . . 10 UHGraph NeighbVtx
4443adantld 474 . . . . . . . . 9 UHGraph NeighbVtx
4544imp 436 . . . . . . . 8 UHGraph NeighbVtx
4645adantld 474 . . . . . . 7 UHGraph NeighbVtx NeighbVtx
4740, 46impbid 195 . . . . . 6 UHGraph NeighbVtx NeighbVtx
48 eleq1 2537 . . . . . . . . 9
4948adantl 473 . . . . . . . 8
50 id 22 . . . . . . . . . 10
51 difeq1 3533 . . . . . . . . . . 11
5251raleqdv 2979 . . . . . . . . . 10 NeighbVtx NeighbVtx
5350, 52raleqbidv 2987 . . . . . . . . 9 NeighbVtx NeighbVtx
54 vex 3034 . . . . . . . . . . 11
55 vex 3034 . . . . . . . . . . 11
56 sneq 3969 . . . . . . . . . . . . 13
5756difeq2d 3540 . . . . . . . . . . . 12
58 oveq2 6316 . . . . . . . . . . . . 13 NeighbVtx NeighbVtx
5958eleq2d 2534 . . . . . . . . . . . 12 NeighbVtx NeighbVtx
6057, 59raleqbidv 2987 . . . . . . . . . . 11 NeighbVtx NeighbVtx
61 sneq 3969 . . . . . . . . . . . . 13
6261difeq2d 3540 . . . . . . . . . . . 12
63 oveq2 6316 . . . . . . . . . . . . 13 NeighbVtx NeighbVtx
6463eleq2d 2534 . . . . . . . . . . . 12 NeighbVtx NeighbVtx
6562, 64raleqbidv 2987 . . . . . . . . . . 11 NeighbVtx NeighbVtx
6654, 55, 60, 65ralpr 4016 . . . . . . . . . 10 NeighbVtx NeighbVtx NeighbVtx
67 difprsn1 4099 . . . . . . . . . . . . 13
6867raleqdv 2979 . . . . . . . . . . . 12 NeighbVtx NeighbVtx
69 eleq1 2537 . . . . . . . . . . . . 13 NeighbVtx NeighbVtx
7055, 69ralsn 4001 . . . . . . . . . . . 12 NeighbVtx NeighbVtx
7168, 70syl6bb 269 . . . . . . . . . . 11 NeighbVtx NeighbVtx
72 difprsn2 4100 . . . . . . . . . . . . 13
7372raleqdv 2979 . . . . . . . . . . . 12 NeighbVtx NeighbVtx
74 eleq1 2537 . . . . . . . . . . . . 13 NeighbVtx NeighbVtx
7554, 74ralsn 4001 . . . . . . . . . . . 12 NeighbVtx NeighbVtx
7673, 75syl6bb 269 . . . . . . . . . . 11 NeighbVtx NeighbVtx
7771, 76anbi12d 725 . . . . . . . . . 10 NeighbVtx NeighbVtx NeighbVtx NeighbVtx
7866, 77syl5bb 265 . . . . . . . . 9 NeighbVtx NeighbVtx NeighbVtx
7953, 78sylan9bbr 715 . . . . . . . 8 NeighbVtx NeighbVtx NeighbVtx
8049, 79bibi12d 328 . . . . . . 7 NeighbVtx NeighbVtx NeighbVtx
8180adantl 473 . . . . . 6 UHGraph NeighbVtx NeighbVtx NeighbVtx
8247, 81mpbird 240 . . . . 5 UHGraph NeighbVtx
8382ex 441 . . . 4 UHGraph NeighbVtx
8483rexlimdvva 2878 . . 3 UHGraph NeighbVtx
855, 84syl5bi 225 . 2 UHGraph NeighbVtx
8685imp 436 1 UHGraph NeighbVtx
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   w3a 1007   wceq 1452   wcel 1904   wne 2641  wral 2756  wrex 2757  cvv 3031   cdif 3387   wss 3390  csn 3959  cpr 3961  cfv 5589  (class class class)co 6308  c2 10681  chash 12553  Vtxcvtx 39251   UHGraph cuhgr 39300  Edgcedga 39371   NeighbVtx cnbgr 39561 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-uhgr 39302  df-edga 39372  df-nbgr 39565 This theorem is referenced by:  uvtx2vtx1edgb  39636
 Copyright terms: Public domain W3C validator