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Theorem 2spot0 24929
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 4563 . . . 4  |-  (/)  e.  _V
2 eleq1 2513 . . . 4  |-  ( V  =  (/)  ->  ( V  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 233 . . 3  |-  ( V  =  (/)  ->  V  e. 
_V )
4 2spthsot 24733 . . 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 24741 . . . . . . . . 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 3772 . . . . . . . . . . 11  |-  ( a  e.  V  ->  -.  V  =  (/) )
87adantr 465 . . . . . . . . . 10  |-  ( ( a  e.  V  /\  b  e.  V )  ->  -.  V  =  (/) )
983ad2ant2 1017 . . . . . . . . 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 731 . . . . . 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 2897 . . . . 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 2897 . . . 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 2855 . . 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 3789 . . 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 2482 1  |-  ( ( V  =  (/)  /\  E  e.  X )  ->  ( V 2SPathOnOt  E )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   E.wrex 2792   {crab 2795   _Vcvv 3093   (/)c0 3767    X. cxp 4983  (class class class)co 6277   2SPathOnOt c2spthot 24721   2SPathOnOt c2pthonot 24722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-2spthonot 24725  df-2spthsot 24726
This theorem is referenced by: (None)
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