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Theorem 2spot0 25269
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 4569 . . . 4  |-  (/)  e.  _V
2 eleq1 2526 . . . 4  |-  ( V  =  (/)  ->  ( V  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 233 . . 3  |-  ( V  =  (/)  ->  V  e. 
_V )
4 2spthsot 25073 . . 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 469 . 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 25081 . . . . . . . . 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 3788 . . . . . . . . . . 11  |-  ( a  e.  V  ->  -.  V  =  (/) )
87adantr 463 . . . . . . . . . 10  |-  ( ( a  e.  V  /\  b  e.  V )  ->  -.  V  =  (/) )
983ad2ant2 1016 . . . . . . . . 9  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( a  e.  V  /\  b  e.  V
)  /\  p  e.  ( ( V  X.  V )  X.  V
) )  ->  -.  V  =  (/) )
106, 9syl 16 . . . . . . . 8  |-  ( p  e.  ( a ( V 2SPathOnOt  E ) b )  ->  -.  V  =  (/) )
1110con2i 120 . . . . . . 7  |-  ( V  =  (/)  ->  -.  p  e.  ( a ( V 2SPathOnOt  E ) b ) )
1211ad4antr 729 . . . . . 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 2910 . . . . 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 2910 . . . 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 2868 . . 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 3806 . . 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 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 ) }  =  (/) )
185, 17eqtrd 2495 1  |-  ( ( V  =  (/)  /\  E  e.  X )  ->  ( V 2SPathOnOt  E )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106   (/)c0 3783    X. cxp 4986  (class class class)co 6270   2SPathOnOt c2spthot 25061   2SPathOnOt c2pthonot 25062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-2spthonot 25065  df-2spthsot 25066
This theorem is referenced by: (None)
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