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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  |-  ( ( V  =  (/)  /\  E  e.  X )  ->  ( V 2SPathOnOt  E )  =  (/) )

Proof of Theorem 2spot0
Dummy variables  a 
b  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4528 . . . 4  |-  (/)  e.  _V
2 eleq1 2537 . . . 4  |-  ( V  =  (/)  ->  ( V  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 241 . . 3  |-  ( V  =  (/)  ->  V  e. 
_V )
4 2spthsot 25675 . . 3  |-  ( ( V  e.  _V  /\  E  e.  X )  ->  ( V 2SPathOnOt  E )  =  { p  e.  ( ( V  X.  V
)  X.  V )  |  E. a  e.  V  E. b  e.  V  p  e.  ( a ( V 2SPathOnOt  E ) b ) } )
53, 4sylan 479 . 2  |-  ( ( V  =  (/)  /\  E  e.  X )  ->  ( V 2SPathOnOt  E )  =  {
p  e.  ( ( V  X.  V )  X.  V )  |  E. a  e.  V  E. b  e.  V  p  e.  ( a
( V 2SPathOnOt  E ) b ) } )
6 2spthonot3v 25683 . . . . . . . . 9  |-  ( p  e.  ( a ( V 2SPathOnOt  E ) b )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  (
a  e.  V  /\  b  e.  V )  /\  p  e.  (
( V  X.  V
)  X.  V ) ) )
7 n0i 3727 . . . . . . . . . . 11  |-  ( a  e.  V  ->  -.  V  =  (/) )
87adantr 472 . . . . . . . . . 10  |-  ( ( a  e.  V  /\  b  e.  V )  ->  -.  V  =  (/) )
983ad2ant2 1052 . . . . . . . . 9  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( a  e.  V  /\  b  e.  V
)  /\  p  e.  ( ( V  X.  V )  X.  V
) )  ->  -.  V  =  (/) )
106, 9syl 17 . . . . . . . 8  |-  ( p  e.  ( a ( V 2SPathOnOt  E ) b )  ->  -.  V  =  (/) )
1110con2i 124 . . . . . . 7  |-  ( V  =  (/)  ->  -.  p  e.  ( a ( V 2SPathOnOt  E ) b ) )
1211ad4antr 746 . . . . . 6  |-  ( ( ( ( ( V  =  (/)  /\  E  e.  X )  /\  p  e.  ( ( V  X.  V )  X.  V
) )  /\  a  e.  V )  /\  b  e.  V )  ->  -.  p  e.  ( a
( V 2SPathOnOt  E ) b ) )
1312nrexdv 2842 . . . . 5  |-  ( ( ( ( V  =  (/)  /\  E  e.  X
)  /\  p  e.  ( ( V  X.  V )  X.  V
) )  /\  a  e.  V )  ->  -.  E. b  e.  V  p  e.  ( a ( V 2SPathOnOt  E ) b ) )
1413nrexdv 2842 . . . 4  |-  ( ( ( V  =  (/)  /\  E  e.  X )  /\  p  e.  ( ( V  X.  V
)  X.  V ) )  ->  -.  E. a  e.  V  E. b  e.  V  p  e.  ( a ( V 2SPathOnOt  E ) b ) )
1514ralrimiva 2809 . . 3  |-  ( ( V  =  (/)  /\  E  e.  X )  ->  A. p  e.  ( ( V  X.  V )  X.  V
)  -.  E. a  e.  V  E. b  e.  V  p  e.  ( a ( V 2SPathOnOt  E ) b ) )
16 rabeq0 3757 . . 3  |-  ( { p  e.  ( ( V  X.  V )  X.  V )  |  E. a  e.  V  E. b  e.  V  p  e.  ( a
( V 2SPathOnOt  E ) b ) }  =  (/)  <->  A. p  e.  ( ( V  X.  V )  X.  V )  -.  E. a  e.  V  E. b  e.  V  p  e.  ( a ( V 2SPathOnOt  E ) b ) )
1715, 16sylibr 217 . 2  |-  ( ( V  =  (/)  /\  E  e.  X )  ->  { p  e.  ( ( V  X.  V )  X.  V
)  |  E. a  e.  V  E. b  e.  V  p  e.  ( a ( V 2SPathOnOt  E ) b ) }  =  (/) )
185, 17eqtrd 2505 1  |-  ( ( V  =  (/)  /\  E  e.  X )  ->  ( V 2SPathOnOt  E )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031   (/)c0 3722    X. 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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