Theorem n4cyclfrgr 27155

Description: There is no 4-cycle in a friendship graph, see Proposition 1(a) of [MertziosUnger] p. 153 : "A friendship graph G contains no C4 as a subgraph ...". (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)

Assertion

Ref	Expression
n4cyclfrgr	|- ( ( G e. FriendGraph /\ F (Cycles ` G ) P ) -> ( # ` F ) =/= 4 )

Proof of Theorem n4cyclfrgr

Dummy variables a b c d k l x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrusgr 27124	. . . . 5 |- ( G e. FriendGraph -> G e. USGraph )
2		usgrupgr 26077	. . . . 5 |- ( G e. USGraph -> G e. UPGraph )
3	1, 2	syl 17	. . . 4 |- ( G e. FriendGraph -> G e. UPGraph )
4		eqid 2622	. . . . . . . . 9 |- (Vtx ` G ) = (Vtx ` G )
5		eqid 2622	. . . . . . . . 9 |- (Edg ` G ) = (Edg ` G )
6	4, 5	upgr4cycl4dv4e 27045	. . . . . . . 8 |- ( ( G e. UPGraph /\ F (Cycles ` G ) P /\ ( # ` F ) = 4 ) -> E. a e. (Vtx ` G ) E. b e. (Vtx ` G ) E. c e. (Vtx ` G ) E. d e. (Vtx ` G ) ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) )
7	4, 5	frgrusgrfrcond 27123	. . . . . . . . . . . 12 |- ( G e. FriendGraph <-> ( G e. USGraph /\ A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) ) )
8		simplrl 800	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> c e. (Vtx ` G ) )
9		necom 2847	. . . . . . . . . . . . . . . . . . . . 21 |- ( a =/= c <-> c =/= a )
10	9	biimpi 206	. . . . . . . . . . . . . . . . . . . 20 |- ( a =/= c -> c =/= a )
11	10	3ad2ant2 1083	. . . . . . . . . . . . . . . . . . 19 |- ( ( a =/= b /\ a =/= c /\ a =/= d ) -> c =/= a )
12	11	ad2antrl 764	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) -> c =/= a )
13	12	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> c =/= a )
14		eldifsn 4317	. . . . . . . . . . . . . . . . 17 |- ( c e. ( (Vtx ` G ) \ { a } ) <-> ( c e. (Vtx ` G ) /\ c =/= a ) )
15	8, 13, 14	sylanbrc 698	. . . . . . . . . . . . . . . 16 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> c e. ( (Vtx ` G ) \ { a } ) )
16		sneq 4187	. . . . . . . . . . . . . . . . . . . 20 |- ( k = a -> { k } = { a } )
17	16	difeq2d 3728	. . . . . . . . . . . . . . . . . . 19 |- ( k = a -> ( (Vtx ` G ) \ { k } ) = ( (Vtx ` G ) \ { a } ) )
18		preq2 4269	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = a -> { x , k } = { x , a } )
19	18	preq1d 4274	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = a -> { { x , k } , { x , l } } = { { x , a } , { x , l } } )
20	19	sseq1d 3632	. . . . . . . . . . . . . . . . . . . 20 |- ( k = a -> ( { { x , k } , { x , l } } C_ (Edg ` G ) <-> { { x , a } , { x , l } } C_ (Edg ` G ) ) )
21	20	reubidv 3126	. . . . . . . . . . . . . . . . . . 19 |- ( k = a -> ( E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) <-> E! x e. (Vtx ` G ) { { x , a } , { x , l } } C_ (Edg ` G ) ) )
22	17, 21	raleqbidv 3152	. . . . . . . . . . . . . . . . . 18 |- ( k = a -> ( A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) <-> A. l e. ( (Vtx ` G ) \ { a } ) E! x e. (Vtx ` G ) { { x , a } , { x , l } } C_ (Edg ` G ) ) )
23	22	rspcv 3305	. . . . . . . . . . . . . . . . 17 |- ( a e. (Vtx ` G ) -> ( A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) -> A. l e. ( (Vtx ` G ) \ { a } ) E! x e. (Vtx ` G ) { { x , a } , { x , l } } C_ (Edg ` G ) ) )
24	23	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) -> A. l e. ( (Vtx ` G ) \ { a } ) E! x e. (Vtx ` G ) { { x , a } , { x , l } } C_ (Edg ` G ) ) )
25		preq2 4269	. . . . . . . . . . . . . . . . . . . 20 |- ( l = c -> { x , l } = { x , c } )
26	25	preq2d 4275	. . . . . . . . . . . . . . . . . . 19 |- ( l = c -> { { x , a } , { x , l } } = { { x , a } , { x , c } } )
27	26	sseq1d 3632	. . . . . . . . . . . . . . . . . 18 |- ( l = c -> ( { { x , a } , { x , l } } C_ (Edg ` G ) <-> { { x , a } , { x , c } } C_ (Edg ` G ) ) )
28	27	reubidv 3126	. . . . . . . . . . . . . . . . 17 |- ( l = c -> ( E! x e. (Vtx ` G ) { { x , a } , { x , l } } C_ (Edg ` G ) <-> E! x e. (Vtx ` G ) { { x , a } , { x , c } } C_ (Edg ` G ) ) )
29	28	rspcv 3305	. . . . . . . . . . . . . . . 16 |- ( c e. ( (Vtx ` G ) \ { a } ) -> ( A. l e. ( (Vtx ` G ) \ { a } ) E! x e. (Vtx ` G ) { { x , a } , { x , l } } C_ (Edg ` G ) -> E! x e. (Vtx ` G ) { { x , a } , { x , c } } C_ (Edg ` G ) ) )
30	15, 24, 29	sylsyld 61	. . . . . . . . . . . . . . 15 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) -> E! x e. (Vtx ` G ) { { x , a } , { x , c } } C_ (Edg ` G ) ) )
31		prcom 4267	. . . . . . . . . . . . . . . . . . 19 |- { x , a } = { a , x }
32	31	preq1i 4271	. . . . . . . . . . . . . . . . . 18 |- { { x , a } , { x , c } } = { { a , x } , { x , c } }
33	32	sseq1i 3629	. . . . . . . . . . . . . . . . 17 |- ( { { x , a } , { x , c } } C_ (Edg ` G ) <-> { { a , x } , { x , c } } C_ (Edg ` G ) )
34	33	reubii 3128	. . . . . . . . . . . . . . . 16 |- ( E! x e. (Vtx ` G ) { { x , a } , { x , c } } C_ (Edg ` G ) <-> E! x e. (Vtx ` G ) { { a , x } , { x , c } } C_ (Edg ` G ) )
35		simpl 473	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) -> ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) )
36	35	ad2antrl 764	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) )
37		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) -> ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) )
38	37	ad2antrl 764	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) )
39		simpllr 799	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> b e. (Vtx ` G ) )
40		simplrr 801	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> d e. (Vtx ` G ) )
41		simprr2 1110	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) -> b =/= d )
42	41	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> b =/= d )
43		4cycl2vnunb 27154	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) /\ ( b e. (Vtx ` G ) /\ d e. (Vtx ` G ) /\ b =/= d ) ) -> -. E! x e. (Vtx ` G ) { { a , x } , { x , c } } C_ (Edg ` G ) )
44	36, 38, 39, 40, 42, 43	syl113anc 1338	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> -. E! x e. (Vtx ` G ) { { a , x } , { x , c } } C_ (Edg ` G ) )
45	44	pm2.21d 118	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( E! x e. (Vtx ` G ) { { a , x } , { x , c } } C_ (Edg ` G ) -> ( # ` F ) =/= 4 ) )
46	45	com12 32	. . . . . . . . . . . . . . . 16 |- ( E! x e. (Vtx ` G ) { { a , x } , { x , c } } C_ (Edg ` G ) -> ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( # ` F ) =/= 4 ) )
47	34, 46	sylbi 207	. . . . . . . . . . . . . . 15 |- ( E! x e. (Vtx ` G ) { { x , a } , { x , c } } C_ (Edg ` G ) -> ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( # ` F ) =/= 4 ) )
48	30, 47	syl6 35	. . . . . . . . . . . . . 14 |- ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) -> ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( # ` F ) =/= 4 ) ) )
49	48	pm2.43b 55	. . . . . . . . . . . . 13 |- ( A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) -> ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( # ` F ) =/= 4 ) )
50	49	adantl 482	. . . . . . . . . . . 12 |- ( ( G e. USGraph /\ A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) ) -> ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( # ` F ) =/= 4 ) )
51	7, 50	sylbi 207	. . . . . . . . . . 11 |- ( G e. FriendGraph -> ( ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) /\ ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( # ` F ) =/= 4 ) )
52	51	expdcom 455	. . . . . . . . . 10 |- ( ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) /\ ( c e. (Vtx ` G ) /\ d e. (Vtx ` G ) ) ) -> ( ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) -> ( G e. FriendGraph -> ( # ` F ) =/= 4 ) ) )
53	52	rexlimdvva 3038	. . . . . . . . 9 |- ( ( a e. (Vtx ` G ) /\ b e. (Vtx ` G ) ) -> ( E. c e. (Vtx ` G ) E. d e. (Vtx ` G ) ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) -> ( G e. FriendGraph -> ( # ` F ) =/= 4 ) ) )
54	53	rexlimivv 3036	. . . . . . . 8 |- ( E. a e. (Vtx ` G ) E. b e. (Vtx ` G ) E. c e. (Vtx ` G ) E. d e. (Vtx ` G ) ( ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) ) /\ ( { c , d } e. (Edg ` G ) /\ { d , a } e. (Edg ` G ) ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) -> ( G e. FriendGraph -> ( # ` F ) =/= 4 ) )
55	6, 54	syl 17	. . . . . . 7 |- ( ( G e. UPGraph /\ F (Cycles ` G ) P /\ ( # ` F ) = 4 ) -> ( G e. FriendGraph -> ( # ` F ) =/= 4 ) )
56	55	3exp 1264	. . . . . 6 |- ( G e. UPGraph -> ( F (Cycles ` G ) P -> ( ( # ` F ) = 4 -> ( G e. FriendGraph -> ( # ` F ) =/= 4 ) ) ) )
57	56	com34 91	. . . . 5 |- ( G e. UPGraph -> ( F (Cycles ` G ) P -> ( G e. FriendGraph -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) ) ) )
58	57	com23 86	. . . 4 |- ( G e. UPGraph -> ( G e. FriendGraph -> ( F (Cycles ` G ) P -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) ) ) )
59	3, 58	mpcom 38	. . 3 |- ( G e. FriendGraph -> ( F (Cycles ` G ) P -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) ) )
60	59	imp 445	. 2 |- ( ( G e. FriendGraph /\ F (Cycles ` G ) P ) -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) )
61		neqne 2802	. 2 |- ( -. ( # ` F ) = 4 -> ( # ` F ) =/= 4 )
62	60, 61	pm2.61d1 171	1 |- ( ( G e. FriendGraph /\ F (Cycles ` G ) P ) -> ( # ` F ) =/= 4 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   class class class wbr 4653   ` cfv 5888   4c4 11072   #chash 13117  Vtxcvtx 25874  Edgcedg 25939   UPGraph cupgr 25975   USGraph cusgr 26044  Cyclesccycls 26680   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-usgr 26046  df-wlks 26495  df-trls 26589  df-pths 26612  df-cycls 26682  df-frgr 27121

This theorem is referenced by: (None)