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Theorem 2spot0 25871
 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 4528 . . . 4
2 eleq1 2537 . . . 4
31, 2mpbiri 241 . . 3
4 2spthsot 25675 . . 3 2SPathOnOt 2SPathOnOt
53, 4sylan 479 . 2 2SPathOnOt 2SPathOnOt
6 2spthonot3v 25683 . . . . . . . . 9 2SPathOnOt
7 n0i 3727 . . . . . . . . . . 11
87adantr 472 . . . . . . . . . 10
983ad2ant2 1052 . . . . . . . . 9
106, 9syl 17 . . . . . . . 8 2SPathOnOt
1110con2i 124 . . . . . . 7 2SPathOnOt
1211ad4antr 746 . . . . . 6 2SPathOnOt
1312nrexdv 2842 . . . . 5 2SPathOnOt
1413nrexdv 2842 . . . 4 2SPathOnOt
1514ralrimiva 2809 . . 3 2SPathOnOt
16 rabeq0 3757 . . 3 2SPathOnOt 2SPathOnOt
1715, 16sylibr 217 . 2 2SPathOnOt
185, 17eqtrd 2505 1 2SPathOnOt
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 376   w3a 1007   wceq 1452   wcel 1904  wral 2756  wrex 2757  crab 2760  cvv 3031  c0 3722   cxp 4837  (class class class)co 6308   2SPathOnOt c2spthot 25663   2SPathOnOt c2pthonot 25664 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 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  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-ral 2761  df-rex 2762  df-reu 2763  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-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  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-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-2spthonot 25667  df-2spthsot 25668 This theorem is referenced by: (None)
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