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Theorem n4cyclfrgra 25825
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.
StepHypRef Expression
1 frisusgra 25799 . . . 4 |- ( V FriendGrph E -> V USGrph E )
2 4cycl4dv4e 25475 . . . . . . . 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 25798 . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . . . . . . 20 |- ( ( c e. V /\ d e. V ) -> c e. V )
54adantl 473 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) -> c e. V )
65adantr 472 . . . . . . . . . . . . . . . . . 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 2696 . . . . . . . . . . . . . . . . . . . . . 22 |- ( a =/= c <-> c =/= a )
87biimpi 199 . . . . . . . . . . . . . . . . . . . . 21 |- ( a =/= c -> c =/= a )
983ad2ant2 1052 . . . . . . . . . . . . . . . . . . . 20 |- ( ( a =/= b /\ a =/= c /\ a =/= d ) -> c =/= a )
109ad2antrl 742 . . . . . . . . . . . . . . . . . . 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 )
1110adantl 473 . . . . . . . . . . . . . . . . . 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 4088 . . . . . . . . . . . . . . . . . 18 |- ( c e. ( V \ { a } ) <-> ( c e. V /\ c =/= a ) )
136, 11, 12sylanbrc 677 . . . . . . . . . . . . . . . . 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 3969 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = a -> { k } = { a } )
1514difeq2d 3540 . . . . . . . . . . . . . . . . . . . . . 22 |- ( k = a -> ( V \ { k } ) = ( V \ { a } ) )
16 preq2 4043 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( k = a -> { x , k } = { x , a } )
1716preq1d 4048 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( k = a -> { { x , k } , { x , l } } = { { x , a } , { x , l } } )
1817sseq1d 3445 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = a -> ( { { x , k } , { x , l } } C_ ran E <-> { { x , a } , { x , l } } C_ ran E ) )
1918reubidv 2961 . . . . . . . . . . . . . . . . . . . . . 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 ) )
2015, 19raleqbidv 2987 . . . . . . . . . . . . . . . . . . . . 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 ) )
2120rspcv 3132 . . . . . . . . . . . . . . . . . . . 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 ) )
2221adantr 472 . . . . . . . . . . . . . . . . . . 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 ) )
2322adantr 472 . . . . . . . . . . . . . . . . . 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 ) )
2423adantr 472 . . . . . . . . . . . . . . . . 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 4043 . . . . . . . . . . . . . . . . . . . . 21 |- ( l = c -> { x , l } = { x , c } )
2625preq2d 4049 . . . . . . . . . . . . . . . . . . . 20 |- ( l = c -> { { x , a } , { x , l } } = { { x , a } , { x , c } } )
2726sseq1d 3445 . . . . . . . . . . . . . . . . . . 19 |- ( l = c -> ( { { x , a } , { x , l } } C_ ran E <-> { { x , a } , { x , c } } C_ ran E ) )
2827reubidv 2961 . . . . . . . . . . . . . . . . . 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 ) )
2928rspcv 3132 . . . . . . . . . . . . . . . . 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 ) )
3013, 24, 29sylsyld 57 . . . . . . . . . . . . . . . 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 4041 . . . . . . . . . . . . . . . . . . . 20 |- { x , a } = { a , x }
3231preq1i 4045 . . . . . . . . . . . . . . . . . . 19 |- { { x , a } , { x , c } } = { { a , x } , { x , c } }
3332sseq1i 3442 . . . . . . . . . . . . . . . . . 18 |- ( { { x , a } , { x , c } } C_ ran E <-> { { a , x } , { x , c } } C_ ran E )
3433reubii 2963 . . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . . . . . . . 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 ) )
3635ad2antrl 742 . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . . 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 ) )
3837ad2antrl 742 . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( a e. V /\ b e. V ) -> b e. V )
4039adantr 472 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) -> b e. V )
4140adantr 472 . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( c e. V /\ d e. V ) -> d e. V )
4342adantl 473 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) -> d e. V )
4443adantr 472 . . . . . . . . . . . . . . . . . . . 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 1079 . . . . . . . . . . . . . . . . . . . . 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 )
4645adantl 473 . . . . . . . . . . . . . . . . . . . 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 25824 . . . . . . . . . . . . . . . . . . . 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 )
4836, 38, 41, 44, 46, 47syl113anc 1304 . . . . . . . . . . . . . . . . . . 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 )
4948pm2.21d 109 . . . . . . . . . . . . . . . . . 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 ) )
5049com12 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 ) )
5134, 50sylbi 200 . . . . . . . . . . . . . . . 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 ) )
5230, 51syl6 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 ) ) )
5352pm2.43b 51 . . . . . . . . . . . . . 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 ) )
5453adantl 473 . . . . . . . . . . . . 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 ) )
553, 54syl 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 ) )
5655com12 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 ) )
5756ex 441 . . . . . . . . . 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 ) ) )
5857rexlimdvva 2878 . . . . . . . . 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 ) ) )
5958rexlimivv 2876 . . . . . . . 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 ) )
602, 59syl 17 . . . . . . 7 |- ( ( V USGrph E /\ F ( V Cycles E ) P /\ ( # ` F ) = 4 ) -> ( V FriendGrph E -> ( # ` F ) =/= 4 ) )
61603exp 1230 . . . . . 6 |- ( V USGrph E -> ( F ( V Cycles E ) P -> ( ( # ` F ) = 4 -> ( V FriendGrph E -> ( # ` F ) =/= 4 ) ) ) )
6261com34 85 . . . . 5 |- ( V USGrph E -> ( F ( V Cycles E ) P -> ( V FriendGrph E -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) ) ) )
6362com23 80 . . . 4 |- ( V USGrph E -> ( V FriendGrph E -> ( F ( V Cycles E ) P -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) ) ) )
641, 63mpcom 36 . . 3 |- ( V FriendGrph E -> ( F ( V Cycles E ) P -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) ) )
6564imp 436 . 2 |- ( ( V FriendGrph E /\ F ( V Cycles E ) P ) -> ( ( # ` F ) = 4 -> ( # ` F ) =/= 4 ) )
66 df-ne 2643 . . 3 |- ( ( # ` F ) =/= 4 <-> -. ( # ` F ) = 4 )
6766biimpri 211 . 2 |- ( -. ( # ` F ) = 4 -> ( # ` F ) =/= 4 )
6865, 67pm2.61d1 164 1 |- ( ( V FriendGrph E /\ F ( V Cycles E ) P ) -> ( # ` F ) =/= 4 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   E.wrex 2757   E!wreu 2758   \ cdif 3387   C_ wss 3390   {csn 3959   {cpr 3961   class class class wbr 4395   ran crn 4840   ` cfv 5589  (class class class)co 6308   4c4 10683   #chash 12553   USGrph cusg 25136   Cycles ccycl 25314   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-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  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-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-usgra 25139  df-wlk 25315  df-trail 25316  df-pth 25317  df-cycl 25320  df-frgra 25796
This theorem is referenced by: (None)
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