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Theorem 2spot0 30792
 Description: If there are no vertices, then there are no paths (of length 2), too. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
Assertion
Ref Expression
2spot0 2SPathOnOt

Proof of Theorem 2spot0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4517 . . . 4
2 eleq1 2521 . . . 4
31, 2mpbiri 233 . . 3
4 2spthsot 30522 . . 3 2SPathOnOt 2SPathOnOt
53, 4sylan 471 . 2 2SPathOnOt 2SPathOnOt
6 2spthonot3v 30530 . . . . . . . . . . 11 2SPathOnOt
7 n0i 3737 . . . . . . . . . . . . 13
87adantr 465 . . . . . . . . . . . 12
983ad2ant2 1010 . . . . . . . . . . 11
106, 9syl 16 . . . . . . . . . 10 2SPathOnOt
1110con2i 120 . . . . . . . . 9 2SPathOnOt
1211ad4antr 731 . . . . . . . 8 2SPathOnOt
1312ralrimiva 2820 . . . . . . 7 2SPathOnOt
14 ralnex 2861 . . . . . . 7 2SPathOnOt 2SPathOnOt
1513, 14sylib 196 . . . . . 6 2SPathOnOt
1615ralrimiva 2820 . . . . 5 2SPathOnOt
17 ralnex 2861 . . . . 5 2SPathOnOt 2SPathOnOt
1816, 17sylib 196 . . . 4 2SPathOnOt
1918ralrimiva 2820 . . 3 2SPathOnOt
20 rabeq0 3754 . . 3 2SPathOnOt 2SPathOnOt
2119, 20sylibr 212 . 2 2SPathOnOt
225, 21eqtrd 2491 1 2SPathOnOt
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 369   w3a 965   wceq 1370   wcel 1758  wral 2793  wrex 2794  crab 2797  cvv 3065  c0 3732   cxp 4933  (class class class)co 6187   2SPathOnOt c2spthot 30510   2SPathOnOt c2pthonot 30511 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-1st 6674  df-2nd 6675  df-2spthonot 30514  df-2spthsot 30515 This theorem is referenced by: (None)
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