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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  |-  ( ( 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 4517 . . . 4  |-  (/)  e.  _V
2 eleq1 2521 . . . 4  |-  ( V  =  (/)  ->  ( V  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 233 . . 3  |-  ( V  =  (/)  ->  V  e. 
_V )
4 2spthsot 30522 . . 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 471 . 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 30530 . . . . . . . . . . 11  |-  ( 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 3737 . . . . . . . . . . . . 13  |-  ( a  e.  V  ->  -.  V  =  (/) )
87adantr 465 . . . . . . . . . . . 12  |-  ( ( a  e.  V  /\  b  e.  V )  ->  -.  V  =  (/) )
983ad2ant2 1010 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( a  e.  V  /\  b  e.  V
)  /\  p  e.  ( ( V  X.  V )  X.  V
) )  ->  -.  V  =  (/) )
106, 9syl 16 . . . . . . . . . 10  |-  ( p  e.  ( a ( V 2SPathOnOt  E ) b )  ->  -.  V  =  (/) )
1110con2i 120 . . . . . . . . 9  |-  ( V  =  (/)  ->  -.  p  e.  ( a ( V 2SPathOnOt  E ) b ) )
1211ad4antr 731 . . . . . . . 8  |-  ( ( ( ( ( V  =  (/)  /\  E  e.  X )  /\  p  e.  ( ( V  X.  V )  X.  V
) )  /\  a  e.  V )  /\  b  e.  V )  ->  -.  p  e.  ( a
( V 2SPathOnOt  E ) b ) )
1312ralrimiva 2820 . . . . . . 7  |-  ( ( ( ( V  =  (/)  /\  E  e.  X
)  /\  p  e.  ( ( V  X.  V )  X.  V
) )  /\  a  e.  V )  ->  A. b  e.  V  -.  p  e.  ( a ( V 2SPathOnOt  E ) b ) )
14 ralnex 2861 . . . . . . 7  |-  ( A. b  e.  V  -.  p  e.  ( a
( V 2SPathOnOt  E ) b )  <->  -.  E. b  e.  V  p  e.  ( a ( V 2SPathOnOt  E ) b ) )
1513, 14sylib 196 . . . . . 6  |-  ( ( ( ( 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 ) )
1615ralrimiva 2820 . . . . 5  |-  ( ( ( V  =  (/)  /\  E  e.  X )  /\  p  e.  ( ( V  X.  V
)  X.  V ) )  ->  A. a  e.  V  -.  E. b  e.  V  p  e.  ( a ( V 2SPathOnOt  E ) b ) )
17 ralnex 2861 . . . . 5  |-  ( A. a  e.  V  -.  E. b  e.  V  p  e.  ( a ( V 2SPathOnOt  E ) b )  <->  -.  E. a  e.  V  E. b  e.  V  p  e.  ( a
( V 2SPathOnOt  E ) b ) )
1816, 17sylib 196 . . . 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 ) )
1918ralrimiva 2820 . . 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 ) )
20 rabeq0 3754 . . 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 ) )
2119, 20sylibr 212 . 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 ) }  =  (/) )
225, 21eqtrd 2491 1  |-  ( ( V  =  (/)  /\  E  e.  X )  ->  ( V 2SPathOnOt  E )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2793   E.wrex 2794   {crab 2797   _Vcvv 3065   (/)c0 3732    X. 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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