Theorem 4cyclusnfrgra 28123

Description: A graph with a 4-cycle is not a friendhip graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.)

Assertion

Ref	Expression
4cyclusnfrgra	|- ( ( V USGrph E /\ ( A e. V /\ C e. V /\ A =/= C ) /\ ( B e. V /\ D e. V /\ B =/= D ) ) -> ( ( ( { A , B } e. ran E /\ { B , C } e. ran E ) /\ ( { C , D } e. ran E /\ { D , A } e. ran E ) ) -> -. V FriendGrph E ) )

Proof of Theorem 4cyclusnfrgra

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 733	. . . . 5 |- ( ( ( V USGrph E /\ ( A e. V /\ C e. V /\ A =/= C ) /\ ( B e. V /\ D e. V /\ B =/= D ) ) /\ ( ( { A , B } e. ran E /\ { B , C } e. ran E ) /\ ( { C , D } e. ran E /\ { D , A } e. ran E ) ) ) -> ( { A , B } e. ran E /\ { B , C } e. ran E ) )
2		simprr 734	. . . . 5 |- ( ( ( V USGrph E /\ ( A e. V /\ C e. V /\ A =/= C ) /\ ( B e. V /\ D e. V /\ B =/= D ) ) /\ ( ( { A , B } e. ran E /\ { B , C } e. ran E ) /\ ( { C , D } e. ran E /\ { D , A } e. ran E ) ) ) -> ( { C , D } e. ran E /\ { D , A } e. ran E ) )
3		simpl3 962	. . . . 5 |- ( ( ( V USGrph E /\ ( A e. V /\ C e. V /\ A =/= C ) /\ ( B e. V /\ D e. V /\ B =/= D ) ) /\ ( ( { A , B } e. ran E /\ { B , C } e. ran E ) /\ ( { C , D } e. ran E /\ { D , A } e. ran E ) ) ) -> ( B e. V /\ D e. V /\ B =/= D ) )
4		4cycl2vnunb 28121	. . . . 5 |- ( ( ( { A , B } e. ran E /\ { B , C } e. ran E ) /\ ( { C , D } e. ran E /\ { D , A } e. ran E ) /\ ( B e. V /\ D e. V /\ B =/= D ) ) -> -. E! x e. V { { A , x } , { x , C } } C_ ran E )
5	1, 2, 3, 4	syl3anc 1184	. . . 4 |- ( ( ( V USGrph E /\ ( A e. V /\ C e. V /\ A =/= C ) /\ ( B e. V /\ D e. V /\ B =/= D ) ) /\ ( ( { A , B } e. ran E /\ { B , C } e. ran E ) /\ ( { C , D } e. ran E /\ { D , A } e. ran E ) ) ) -> -. E! x e. V { { A , x } , { x , C } } C_ ran E )
6		frgraunss 28099	. . . . . . . 8 |- ( V FriendGrph E -> ( ( A e. V /\ C e. V /\ A =/= C ) -> E! x e. V { { A , x } , { x , C } } C_ ran E ) )
7		pm2.24 103	. . . . . . . 8 |- ( E! x e. V { { A , x } , { x , C } } C_ ran E -> ( -. E! x e. V { { A , x } , { x , C } } C_ ran E -> -. V FriendGrph E ) )
8	6, 7	syl6com 33	. . . . . . 7 |- ( ( A e. V /\ C e. V /\ A =/= C ) -> ( V FriendGrph E -> ( -. E! x e. V { { A , x } , { x , C } } C_ ran E -> -. V FriendGrph E ) ) )
9	8	3ad2ant2 979	. . . . . 6 |- ( ( V USGrph E /\ ( A e. V /\ C e. V /\ A =/= C ) /\ ( B e. V /\ D e. V /\ B =/= D ) ) -> ( V FriendGrph E -> ( -. E! x e. V { { A , x } , { x , C } } C_ ran E -> -. V FriendGrph E ) ) )
10	9	com23 74	. . . . 5 |- ( ( V USGrph E /\ ( A e. V /\ C e. V /\ A =/= C ) /\ ( B e. V /\ D e. V /\ B =/= D ) ) -> ( -. E! x e. V { { A , x } , { x , C } } C_ ran E -> ( V FriendGrph E -> -. V FriendGrph E ) ) )
11	10	adantr 452	. . . 4 |- ( ( ( V USGrph E /\ ( A e. V /\ C e. V /\ A =/= C ) /\ ( B e. V /\ D e. V /\ B =/= D ) ) /\ ( ( { A , B } e. ran E /\ { B , C } e. ran E ) /\ ( { C , D } e. ran E /\ { D , A } e. ran E ) ) ) -> ( -. E! x e. V { { A , x } , { x , C } } C_ ran E -> ( V FriendGrph E -> -. V FriendGrph E ) ) )
12	5, 11	mpd 15	. . 3 |- ( ( ( V USGrph E /\ ( A e. V /\ C e. V /\ A =/= C ) /\ ( B e. V /\ D e. V /\ B =/= D ) ) /\ ( ( { A , B } e. ran E /\ { B , C } e. ran E ) /\ ( { C , D } e. ran E /\ { D , A } e. ran E ) ) ) -> ( V FriendGrph E -> -. V FriendGrph E ) )
13	12	pm2.01d 163	. 2 |- ( ( ( V USGrph E /\ ( A e. V /\ C e. V /\ A =/= C ) /\ ( B e. V /\ D e. V /\ B =/= D ) ) /\ ( ( { A , B } e. ran E /\ { B , C } e. ran E ) /\ ( { C , D } e. ran E /\ { D , A } e. ran E ) ) ) -> -. V FriendGrph E )
14	13	ex 424	1 |- ( ( V USGrph E /\ ( A e. V /\ C e. V /\ A =/= C ) /\ ( B e. V /\ D e. V /\ B =/= D ) ) -> ( ( ( { A , B } e. ran E /\ { B , C } e. ran E ) /\ ( { C , D } e. ran E /\ { D , A } e. ran E ) ) -> -. V FriendGrph E ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   /\ w3a 936   e. wcel 1721   =/= wne 2567   E!wreu 2668   C_ wss 3280   {cpr 3775   class class class wbr 4172   ran crn 4838   USGrph cusg 21318   FriendGrph cfrgra 28092

This theorem is referenced by:  frgranbnb  28124  frgrawopreg  28152

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-frgra 28093