Theorem 4cyclusnfrgra 25826

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 772	. . . . 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 774	. . . . 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 1035	. . . . 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 25824	. . . . 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 1292	. . . 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 25802	. . . . . . . 8 |- ( V FriendGrph E -> ( ( A e. V /\ C e. V /\ A =/= C ) -> E! x e. V { { A , x } , { x , C } } C_ ran E ) )
7		pm2.24 112	. . . . . . . 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 35	. . . . . . 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 1052	. . . . . 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 80	. . . . 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 472	. . . 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 174	. 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 441	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 376   /\ w3a 1007   e. wcel 1904   =/= wne 2641   E!wreu 2758   C_ wss 3390   {cpr 3961   class class class wbr 4395   ran crn 4840   USGrph cusg 25136   FriendGrph cfrgra 25795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850  df-frgra 25796

This theorem is referenced by:  frgranbnb  25827  frgrawopreg  25856