Theorem n4cyclfrgra 25144

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.)

Assertion

Ref	Expression
n4cyclfrgra	|- ( ( V FriendGrph E /\ F ( V Cycles E ) P ) -> ( # ` F ) =/= 4 )

Proof of Theorem n4cyclfrgra

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

Step	Hyp	Ref	Expression
1		frisusgra 25118	. . . 4 |- ( V FriendGrph E -> V USGrph E )
2		4cycl4dv4e 24794	. . . . . . . 8 |- ( ( V USGrph E /\ F ( V Cycles E ) P /\ ( # ` F ) = 4 ) -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) )
3		frisusgrapr 25117	. . . . . . . . . . . . 13 |- ( V FriendGrph E -> ( V USGrph E /\ A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ ran E ) )
4		simpl 457	. . . . . . . . . . . . . . . . . . . 20 |- ( ( c e. V /\ d e. V ) -> c e. V )
5	4	adantl 466	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) -> c e. V )
6	5	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> c e. V )
7		necom 2726	. . . . . . . . . . . . . . . . . . . . . 22 |- ( a =/= c <-> c =/= a )
8	7	biimpi 194	. . . . . . . . . . . . . . . . . . . . 21 |- ( a =/= c -> c =/= a )
9	8	3ad2ant2 1018	. . . . . . . . . . . . . . . . . . . 20 |- ( ( a =/= b /\ a =/= c /\ a =/= d ) -> c =/= a )
10	9	ad2antrl 727	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) -> c =/= a )
11	10	adantl 466	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> c =/= a )
12		eldifsn 4157	. . . . . . . . . . . . . . . . . 18 |- ( c e. ( V \ { a } ) <-> ( c e. V /\ c =/= a ) )
13	6, 11, 12	sylanbrc 664	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> c e. ( V \ { a } ) )
14		sneq 4042	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = a -> { k } = { a } )
15	14	difeq2d 3618	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = a -> ( V \ { k } ) = ( V \ { a } ) )
16		preq2 4112	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( k = a -> { x , k } = { x , a } )
17	16	preq1d 4117	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( k = a -> { { x , k } , { x , l } } = { { x , a } , { x , l } } )
18	17	sseq1d 3526	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = a -> ( { { x , k } , { x , l } } C_ ran E <-> { { x , a } , { x , l } } C_ ran E ) )
19	18	reubidv 3042	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = a -> ( E! x e. V { { x , k } , { x , l } } C_ ran E <-> E! x e. V { { x , a } , { x , l } } C_ ran E ) )
20	15, 19	raleqbidv 3068	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = a -> ( A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ ran E <-> A. l e. ( V \ { a } ) E! x e. V { { x , a } , { x , l } } C_ ran E ) )
21	20	rspcv 3206	. . . . . . . . . . . . . . . . . . . 20 |- ( a e. V -> ( A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ ran E -> A. l e. ( V \ { a } ) E! x e. V { { x , a } , { x , l } } C_ ran E ) )
22	21	adantr 465	. . . . . . . . . . . . . . . . . . 19 |- ( ( a e. V /\ b e. V ) -> ( A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ ran E -> A. l e. ( V \ { a } ) E! x e. V { { x , a } , { x , l } } C_ ran E ) )
23	22	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) -> ( A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ ran E -> A. l e. ( V \ { a } ) E! x e. V { { x , a } , { x , l } } C_ ran E ) )
24	23	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ ran E -> A. l e. ( V \ { a } ) E! x e. V { { x , a } , { x , l } } C_ ran E ) )
25		preq2 4112	. . . . . . . . . . . . . . . . . . . . 21 |- ( l = c -> { x , l } = { x , c } )
26	25	preq2d 4118	. . . . . . . . . . . . . . . . . . . 20 |- ( l = c -> { { x , a } , { x , l } } = { { x , a } , { x , c } } )
27	26	sseq1d 3526	. . . . . . . . . . . . . . . . . . 19 |- ( l = c -> ( { { x , a } , { x , l } } C_ ran E <-> { { x , a } , { x , c } } C_ ran E ) )
28	27	reubidv 3042	. . . . . . . . . . . . . . . . . 18 |- ( l = c -> ( E! x e. V { { x , a } , { x , l } } C_ ran E <-> E! x e. V { { x , a } , { x , c } } C_ ran E ) )
29	28	rspcv 3206	. . . . . . . . . . . . . . . . 17 |- ( c e. ( V \ { a } ) -> ( A. l e. ( V \ { a } ) E! x e. V { { x , a } , { x , l } } C_ ran E -> E! x e. V { { x , a } , { x , c } } C_ ran E ) )
30	13, 24, 29	sylsyld 56	. . . . . . . . . . . . . . . 16 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ ran E -> E! x e. V { { x , a } , { x , c } } C_ ran E ) )
31		prcom 4110	. . . . . . . . . . . . . . . . . . . 20 |- { x , a } = { a , x }
32	31	preq1i 4114	. . . . . . . . . . . . . . . . . . 19 |- { { x , a } , { x , c } } = { { a , x } , { x , c } }
33	32	sseq1i 3523	. . . . . . . . . . . . . . . . . 18 |- ( { { x , a } , { x , c } } C_ ran E <-> { { a , x } , { x , c } } C_ ran E )
34	33	reubii 3044	. . . . . . . . . . . . . . . . 17 |- ( E! x e. V { { x , a } , { x , c } } C_ ran E <-> E! x e. V { { a , x } , { x , c } } C_ ran E )
35		simpl 457	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( { 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 ) )
36	35	ad2antrl 727	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( { a , b } e. ran E /\ { b , c } e. ran E ) )
37		simpr 461	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( { 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 ) )
38	37	ad2antrl 727	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( { c , d } e. ran E /\ { d , a } e. ran E ) )
39		simpr 461	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( a e. V /\ b e. V ) -> b e. V )
40	39	adantr 465	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) -> b e. V )
41	40	adantr 465	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> b e. V )
42		simpr 461	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( c e. V /\ d e. V ) -> d e. V )
43	42	adantl 466	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) -> d e. V )
44	43	adantr 465	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> d e. V )
45		simprr2 1045	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) -> b =/= d )
46	45	adantl 466	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> b =/= d )
47		4cycl2vnunb 25143	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( { 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 )
48	36, 38, 41, 44, 46, 47	syl113anc 1240	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> -. E! x e. V { { a , x } , { x , c } } C_ ran E )
49	48	pm2.21d 106	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( E! x e. V { { a , x } , { x , c } } C_ ran E -> ( # ` F ) =/= 4 ) )
50	49	com12 31	. . . . . . . . . . . . . . . . 17 |- ( E! x e. V { { a , x } , { x , c } } C_ ran E -> ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( # ` F ) =/= 4 ) )
51	34, 50	sylbi 195	. . . . . . . . . . . . . . . 16 |- ( E! x e. V { { x , a } , { x , c } } C_ ran E -> ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( # ` F ) =/= 4 ) )
52	30, 51	syl6 33	. . . . . . . . . . . . . . 15 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ ran E -> ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( # ` F ) =/= 4 ) ) )
53	52	pm2.43b 50	. . . . . . . . . . . . . 14 |- ( A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ ran E -> ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( # ` F ) =/= 4 ) )
54	53	adantl 466	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ ran E ) -> ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( # ` F ) =/= 4 ) )
55	3, 54	syl 16	. . . . . . . . . . . 12 |- ( V FriendGrph E -> ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( # ` F ) =/= 4 ) )
56	55	com12 31	. . . . . . . . . . 11 |- ( ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) /\ ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) -> ( V FriendGrph E -> ( # ` F ) =/= 4 ) )
57	56	ex 434	. . . . . . . . . 10 |- ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) -> ( ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) -> ( V FriendGrph E -> ( # ` F ) =/= 4 ) ) )
58	57	rexlimdvva 2956	. . . . . . . . 9 |- ( ( a e. V /\ b e. V ) -> ( E. c e. V E. d e. V ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) -> ( V FriendGrph E -> ( # ` F ) =/= 4 ) ) )
59	58	rexlimivv 2954	. . . . . . . 8 |- ( E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. ran E /\ { b , c } e. ran E ) /\ ( { c , d } e. ran E /\ { d , a } e. ran E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) -> ( V FriendGrph E -> ( # ` F ) =/= 4 ) )
60	2, 59	syl 16	. . . . . . 7 |- ( ( V USGrph E /\ F ( V Cycles E ) P /\ ( # ` F ) = 4 ) -> ( V FriendGrph E -> ( # ` F ) =/= 4 ) )
61	60	3exp 1195	. . . . . 6 |- ( V USGrph E -> ( F ( V Cycles E ) P -> ( ( # ` F ) = 4 -> ( V FriendGrph E -> ( # ` F ) =/= 4 ) ) ) )
62	61	com34 83	. . . . 5 |- ( V USGrph E -> ( F ( V Cycles E ) P -> ( V FriendGrph E -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) ) ) )
63	62	com23 78	. . . 4 |- ( V USGrph E -> ( V FriendGrph E -> ( F ( V Cycles E ) P -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) ) ) )
64	1, 63	mpcom 36	. . 3 |- ( V FriendGrph E -> ( F ( V Cycles E ) P -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) ) )
65	64	imp 429	. 2 |- ( ( V FriendGrph E /\ F ( V Cycles E ) P ) -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) )
66		df-ne 2654	. . 3 |- ( ( # ` F ) =/= 4 <-> -. ( # ` F ) = 4 )
67	66	biimpri 206	. 2 |- ( -. ( # ` F ) = 4 -> ( # ` F ) =/= 4 )
68	65, 67	pm2.61d1 159	1 |- ( ( V FriendGrph E /\ F ( V Cycles E ) P ) -> ( # ` F ) =/= 4 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   A.wral 2807   E.wrex 2808   E!wreu 2809   \ cdif 3468   C_ wss 3471   {csn 4032   {cpr 4034   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296   4c4 10608   #chash 12407   USGrph cusg 24456   Cycles ccycl 24633   FriendGrph cfrgra 25114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-hash 12408  df-word 12545  df-usgra 24459  df-wlk 24634  df-trail 24635  df-pth 24636  df-cycl 24639  df-frgra 25115

This theorem is referenced by: (None)