Theorem 4cyclusnfrgra 30616

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 755	. . . . 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 756	. . . . 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 993	. . . . 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 30614	. . . . 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 1218	. . . 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 30592	. . . . . . . 8 |- ( V FriendGrph E -> ( ( A e. V /\ C e. V /\ A =/= C ) -> E! x e. V { { A , x } , { x , C } } C_ ran E ) )
7		pm2.24 109	. . . . . . . 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 1010	. . . . . 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 78	. . . . 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 465	. . . 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 169	. 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 434	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 369   /\ w3a 965   e. wcel 1756   =/= wne 2611   E!wreu 2722   C_ wss 3333   {cpr 3884   class class class wbr 4297   ran crn 4846   USGrph cusg 23269   FriendGrph cfrgra 30585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-frgra 30586

This theorem is referenced by:  frgranbnb  30617  frgrawopreg  30647