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Theorem 4cycl2v2nb 28407
Description: In a (maybe degenerated) 4-cycle, two vertices have two (maybe not different) common neighbors. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
Assertion
Ref Expression
4cycl2v2nb  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  ->  ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E ) )

Proof of Theorem 4cycl2v2nb
StepHypRef Expression
1 prssi 3955 . 2  |-  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  { { A ,  B } ,  { B ,  C } }  C_  ran  E )
2 prcom 3883 . . . . 5  |-  { D ,  A }  =  { A ,  D }
32eleq1i 2500 . . . 4  |-  ( { D ,  A }  e.  ran  E  <->  { A ,  D }  e.  ran  E )
43biimpi 188 . . 3  |-  ( { D ,  A }  e.  ran  E  ->  { A ,  D }  e.  ran  E )
5 prcom 3883 . . . . 5  |-  { C ,  D }  =  { D ,  C }
65eleq1i 2500 . . . 4  |-  ( { C ,  D }  e.  ran  E  <->  { D ,  C }  e.  ran  E )
76biimpi 188 . . 3  |-  ( { C ,  D }  e.  ran  E  ->  { D ,  C }  e.  ran  E )
8 prssi 3955 . . 3  |-  ( ( { A ,  D }  e.  ran  E  /\  { D ,  C }  e.  ran  E )  ->  { { A ,  D } ,  { D ,  C } }  C_  ran  E )
94, 7, 8syl2anr 466 . 2  |-  ( ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E )  ->  { { A ,  D } ,  { D ,  C } }  C_  ran  E )
101, 9anim12i 551 1  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  ->  ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726    C_ wss 3321   {cpr 3816   ran crn 4880
This theorem is referenced by:  4cycl2vnunb  28408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-un 3326  df-in 3328  df-ss 3335  df-sn 3821  df-pr 3822
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