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Theorem 4cycl2v2nb 28438
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 3787 . 2  |-  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  { { A ,  B } ,  { B ,  C } }  C_  ran  E )
2 prcom 3718 . . . . 5  |-  { D ,  A }  =  { A ,  D }
32eleq1i 2359 . . . 4  |-  ( { D ,  A }  e.  ran  E  <->  { A ,  D }  e.  ran  E )
43biimpi 186 . . 3  |-  ( { D ,  A }  e.  ran  E  ->  { A ,  D }  e.  ran  E )
5 prcom 3718 . . . . 5  |-  { C ,  D }  =  { D ,  C }
65eleq1i 2359 . . . 4  |-  ( { C ,  D }  e.  ran  E  <->  { D ,  C }  e.  ran  E )
76biimpi 186 . . 3  |-  ( { C ,  D }  e.  ran  E  ->  { D ,  C }  e.  ran  E )
8 prssi 3787 . . 3  |-  ( ( { A ,  D }  e.  ran  E  /\  { D ,  C }  e.  ran  E )  ->  { { A ,  D } ,  { D ,  C } }  C_  ran  E )
94, 7, 8syl2anr 464 . 2  |-  ( ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E )  ->  { { A ,  D } ,  { D ,  C } }  C_  ran  E )
101, 9anim12i 549 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 358    e. wcel 1696    C_ wss 3165   {cpr 3654   ran crn 4706
This theorem is referenced by:  4cycl2vnunb  28439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-sn 3659  df-pr 3660
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