Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nbgr2vtx1edg Structured version   Visualization version   Unicode version

Theorem nbgr2vtx1edg 39582
 Description: If a graph 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.) (Revised by AV, 25-Mar-2021.)
Hypotheses
Ref Expression
nbgr2vtx1edg.v Vtx
nbgr2vtx1edg.e Edg
Assertion
Ref Expression
nbgr2vtx1edg NeighbVtx
Distinct variable groups:   ,   ,,   ,,
Allowed substitution hint:   ()

Proof of Theorem nbgr2vtx1edg
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 simpll 768 . . . . . . . . . 10
76ancomd 458 . . . . . . . . 9
8 simpl 464 . . . . . . . . . . 11
98necomd 2698 . . . . . . . . . 10
109ad2antlr 741 . . . . . . . . 9
11 id 22 . . . . . . . . . . 11
12 sseq2 3440 . . . . . . . . . . . 12
1312adantl 473 . . . . . . . . . . 11
14 ssid 3437 . . . . . . . . . . . 12
1514a1i 11 . . . . . . . . . . 11
1611, 13, 15rspcedvd 3143 . . . . . . . . . 10
1716adantl 473 . . . . . . . . 9
1811vgrex 39257 . . . . . . . . . . 11
19 nbgr2vtx1edg.e . . . . . . . . . . . 12 Edg
201, 19nbgrel 39574 . . . . . . . . . . 11 NeighbVtx
2118, 20syl 17 . . . . . . . . . 10 NeighbVtx
2221ad3antrrr 744 . . . . . . . . 9 NeighbVtx
237, 10, 17, 22mpbir3and 1213 . . . . . . . 8 NeighbVtx
248ad2antlr 741 . . . . . . . . 9
25 sseq2 3440 . . . . . . . . . . . 12
2625adantl 473 . . . . . . . . . . 11
27 prcom 4041 . . . . . . . . . . . . 13
2827eqimssi 3472 . . . . . . . . . . . 12
2928a1i 11 . . . . . . . . . . 11
3011, 26, 29rspcedvd 3143 . . . . . . . . . 10
3130adantl 473 . . . . . . . . 9
321, 19nbgrel 39574 . . . . . . . . . . 11 NeighbVtx
3318, 32syl 17 . . . . . . . . . 10 NeighbVtx
3433ad3antrrr 744 . . . . . . . . 9 NeighbVtx
356, 24, 31, 34mpbir3and 1213 . . . . . . . 8 NeighbVtx
36 difprsn1 4099 . . . . . . . . . . . . 13
3736raleqdv 2979 . . . . . . . . . . . 12 NeighbVtx NeighbVtx
38 vex 3034 . . . . . . . . . . . . 13
39 eleq1 2537 . . . . . . . . . . . . 13 NeighbVtx NeighbVtx
4038, 39ralsn 4001 . . . . . . . . . . . 12 NeighbVtx NeighbVtx
4137, 40syl6bb 269 . . . . . . . . . . 11 NeighbVtx NeighbVtx
42 difprsn2 4100 . . . . . . . . . . . . 13
4342raleqdv 2979 . . . . . . . . . . . 12 NeighbVtx NeighbVtx
44 vex 3034 . . . . . . . . . . . . 13
45 eleq1 2537 . . . . . . . . . . . . 13 NeighbVtx NeighbVtx
4644, 45ralsn 4001 . . . . . . . . . . . 12 NeighbVtx NeighbVtx
4743, 46syl6bb 269 . . . . . . . . . . 11 NeighbVtx NeighbVtx
4841, 47anbi12d 725 . . . . . . . . . 10 NeighbVtx NeighbVtx NeighbVtx NeighbVtx
4948adantr 472 . . . . . . . . 9 NeighbVtx NeighbVtx NeighbVtx NeighbVtx
5049ad2antlr 741 . . . . . . . 8 NeighbVtx NeighbVtx NeighbVtx NeighbVtx
5123, 35, 50mpbir2and 936 . . . . . . 7 NeighbVtx NeighbVtx
5251ex 441 . . . . . 6 NeighbVtx NeighbVtx
53 eleq1 2537 . . . . . . . . 9
54 id 22 . . . . . . . . . . 11
55 difeq1 3533 . . . . . . . . . . . 12
5655raleqdv 2979 . . . . . . . . . . 11 NeighbVtx NeighbVtx
5754, 56raleqbidv 2987 . . . . . . . . . 10 NeighbVtx NeighbVtx
58 sneq 3969 . . . . . . . . . . . . 13
5958difeq2d 3540 . . . . . . . . . . . 12
60 oveq2 6316 . . . . . . . . . . . . 13 NeighbVtx NeighbVtx
6160eleq2d 2534 . . . . . . . . . . . 12 NeighbVtx NeighbVtx
6259, 61raleqbidv 2987 . . . . . . . . . . 11 NeighbVtx NeighbVtx
63 sneq 3969 . . . . . . . . . . . . 13
6463difeq2d 3540 . . . . . . . . . . . 12
65 oveq2 6316 . . . . . . . . . . . . 13 NeighbVtx NeighbVtx
6665eleq2d 2534 . . . . . . . . . . . 12 NeighbVtx NeighbVtx
6764, 66raleqbidv 2987 . . . . . . . . . . 11 NeighbVtx NeighbVtx
6844, 38, 62, 67ralpr 4016 . . . . . . . . . 10 NeighbVtx NeighbVtx NeighbVtx
6957, 68syl6bb 269 . . . . . . . . 9 NeighbVtx NeighbVtx NeighbVtx
7053, 69imbi12d 327 . . . . . . . 8 NeighbVtx NeighbVtx NeighbVtx
7170adantl 473 . . . . . . 7 NeighbVtx NeighbVtx NeighbVtx
7271adantl 473 . . . . . 6 NeighbVtx NeighbVtx NeighbVtx
7352, 72mpbird 240 . . . . 5 NeighbVtx
7473ex 441 . . . 4 NeighbVtx
7574rexlimivv 2876 . . 3 NeighbVtx
765, 75sylbi 200 . 2 NeighbVtx
7776imp 436 1 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  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-nbgr 39565 This theorem is referenced by:  uvtx2vtx1edg  39635
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