Theorem n4cyclfrgra 25746

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 25720	. . . 4 |- ( V FriendGrph E -> V USGrph E )
2		4cycl4dv4e 25396	. . . . . . . 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 25719	. . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . . . . 20 |- ( ( c e. V /\ d e. V ) -> c e. V )
5	4	adantl 468	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) -> c e. V )
6	5	adantr 467	. . . . . . . . . . . . . . . . . 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 2677	. . . . . . . . . . . . . . . . . . . . . 22 |- ( a =/= c <-> c =/= a )
8	7	biimpi 198	. . . . . . . . . . . . . . . . . . . . 21 |- ( a =/= c -> c =/= a )
9	8	3ad2ant2 1030	. . . . . . . . . . . . . . . . . . . 20 |- ( ( a =/= b /\ a =/= c /\ a =/= d ) -> c =/= a )
10	9	ad2antrl 734	. . . . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . . . 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 4097	. . . . . . . . . . . . . . . . . 18 |- ( c e. ( V \ { a } ) <-> ( c e. V /\ c =/= a ) )
13	6, 11, 12	sylanbrc 670	. . . . . . . . . . . . . . . . 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 3978	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = a -> { k } = { a } )
15	14	difeq2d 3551	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = a -> ( V \ { k } ) = ( V \ { a } ) )
16		preq2 4052	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( k = a -> { x , k } = { x , a } )
17	16	preq1d 4057	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( k = a -> { { x , k } , { x , l } } = { { x , a } , { x , l } } )
18	17	sseq1d 3459	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = a -> ( { { x , k } , { x , l } } C_ ran E <-> { { x , a } , { x , l } } C_ ran E ) )
19	18	reubidv 2975	. . . . . . . . . . . . . . . . . . . . . 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 3001	. . . . . . . . . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . 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 4052	. . . . . . . . . . . . . . . . . . . . 21 |- ( l = c -> { x , l } = { x , c } )
26	25	preq2d 4058	. . . . . . . . . . . . . . . . . . . 20 |- ( l = c -> { { x , a } , { x , l } } = { { x , a } , { x , c } } )
27	26	sseq1d 3459	. . . . . . . . . . . . . . . . . . 19 |- ( l = c -> ( { { x , a } , { x , l } } C_ ran E <-> { { x , a } , { x , c } } C_ ran E ) )
28	27	reubidv 2975	. . . . . . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . . . . . 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 58	. . . . . . . . . . . . . . . 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 4050	. . . . . . . . . . . . . . . . . . . 20 |- { x , a } = { a , x }
32	31	preq1i 4054	. . . . . . . . . . . . . . . . . . 19 |- { { x , a } , { x , c } } = { { a , x } , { x , c } }
33	32	sseq1i 3456	. . . . . . . . . . . . . . . . . 18 |- ( { { x , a } , { x , c } } C_ ran E <-> { { a , x } , { x , c } } C_ ran E )
34	33	reubii 2977	. . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( a e. V /\ b e. V ) -> b e. V )
40	39	adantr 467	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) -> b e. V )
41	40	adantr 467	. . . . . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( c e. V /\ d e. V ) -> d e. V )
43	42	adantl 468	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) -> d e. V )
44	43	adantr 467	. . . . . . . . . . . . . . . . . . . 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 1057	. . . . . . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . . . . . 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 25745	. . . . . . . . . . . . . . . . . . . 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 1280	. . . . . . . . . . . . . . . . . . 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 110	. . . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . . 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 199	. . . . . . . . . . . . . . . 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 34	. . . . . . . . . . . . . . 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 52	. . . . . . . . . . . . . 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 468	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 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 32	. . . . . . . . . . 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 436	. . . . . . . . . 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 2886	. . . . . . . . 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 2884	. . . . . . . 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 17	. . . . . . 7 |- ( ( V USGrph E /\ F ( V Cycles E ) P /\ ( # ` F ) = 4 ) -> ( V FriendGrph E -> ( # ` F ) =/= 4 ) )
61	60	3exp 1207	. . . . . 6 |- ( V USGrph E -> ( F ( V Cycles E ) P -> ( ( # ` F ) = 4 -> ( V FriendGrph E -> ( # ` F ) =/= 4 ) ) ) )
62	61	com34 86	. . . . 5 |- ( V USGrph E -> ( F ( V Cycles E ) P -> ( V FriendGrph E -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) ) ) )
63	62	com23 81	. . . 4 |- ( V USGrph E -> ( V FriendGrph E -> ( F ( V Cycles E ) P -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) ) ) )
64	1, 63	mpcom 37	. . 3 |- ( V FriendGrph E -> ( F ( V Cycles E ) P -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) ) )
65	64	imp 431	. 2 |- ( ( V FriendGrph E /\ F ( V Cycles E ) P ) -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) )
66		df-ne 2624	. . 3 |- ( ( # ` F ) =/= 4 <-> -. ( # ` F ) = 4 )
67	66	biimpri 210	. 2 |- ( -. ( # ` F ) = 4 -> ( # ` F ) =/= 4 )
68	65, 67	pm2.61d1 163	1 |- ( ( V FriendGrph E /\ F ( V Cycles E ) P ) -> ( # ` F ) =/= 4 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   E!wreu 2739   \ cdif 3401   C_ wss 3404   {csn 3968   {cpr 3970   class class class wbr 4402   ran crn 4835   ` cfv 5582  (class class class)co 6290   4c4 10661   #chash 12515   USGrph cusg 25057   Cycles ccycl 25235   FriendGrph cfrgra 25716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-usgra 25060  df-wlk 25236  df-trail 25237  df-pth 25238  df-cycl 25241  df-frgra 25717

This theorem is referenced by: (None)