Theorem usgrexmpldifpr 16068

Description: Lemma for usgrexmpledg : all "edges" are different. (Contributed by Alexander van der Vekens, 15-Aug-2017.)

Assertion

Ref	Expression
usgrexmpldifpr	|- ( ( { 0 , 1 } =/= { 1 , 2 } /\ { 0 , 1 } =/= { 2 , 0 } /\ { 0 , 1 } =/= { 0 , 3 } ) /\ ( { 1 , 2 } =/= { 2 , 0 } /\ { 1 , 2 } =/= { 0 , 3 } /\ { 2 , 0 } =/= { 0 , 3 } ) )

Proof of Theorem usgrexmpldifpr

Step	Hyp	Ref	Expression
1		0z 9473	. . . . . 6 |- 0 e. ZZ
2		1z 9488	. . . . . 6 |- 1 e. ZZ
3	1, 2	pm3.2i 272	. . . . 5 |- ( 0 e. ZZ /\ 1 e. ZZ )
4		2z 9490	. . . . . 6 |- 2 e. ZZ
5	2, 4	pm3.2i 272	. . . . 5 |- ( 1 e. ZZ /\ 2 e. ZZ )
6	3, 5	pm3.2i 272	. . . 4 |- ( ( 0 e. ZZ /\ 1 e. ZZ ) /\ ( 1 e. ZZ /\ 2 e. ZZ ) )
7		1ne0 9194	. . . . . . 7 |- 1 =/= 0
8	7	necomi 2485	. . . . . 6 |- 0 =/= 1
9		2ne0 9218	. . . . . . 7 |- 2 =/= 0
10	9	necomi 2485	. . . . . 6 |- 0 =/= 2
11	8, 10	pm3.2i 272	. . . . 5 |- ( 0 =/= 1 /\ 0 =/= 2 )
12	11	orci 736	. . . 4 |- ( ( 0 =/= 1 /\ 0 =/= 2 ) \/ ( 1 =/= 1 /\ 1 =/= 2 ) )
13		prneimg 3852	. . . 4 |- ( ( ( 0 e. ZZ /\ 1 e. ZZ ) /\ ( 1 e. ZZ /\ 2 e. ZZ ) ) -> ( ( ( 0 =/= 1 /\ 0 =/= 2 ) \/ ( 1 =/= 1 /\ 1 =/= 2 ) ) -> { 0 , 1 } =/= { 1 , 2 } ) )
14	6, 12, 13	mp2 16	. . 3 |- { 0 , 1 } =/= { 1 , 2 }
15	4, 1	pm3.2i 272	. . . . 5 |- ( 2 e. ZZ /\ 0 e. ZZ )
16	3, 15	pm3.2i 272	. . . 4 |- ( ( 0 e. ZZ /\ 1 e. ZZ ) /\ ( 2 e. ZZ /\ 0 e. ZZ ) )
17		1ne2 9333	. . . . . 6 |- 1 =/= 2
18	17, 7	pm3.2i 272	. . . . 5 |- ( 1 =/= 2 /\ 1 =/= 0 )
19	18	olci 737	. . . 4 |- ( ( 0 =/= 2 /\ 0 =/= 0 ) \/ ( 1 =/= 2 /\ 1 =/= 0 ) )
20		prneimg 3852	. . . 4 |- ( ( ( 0 e. ZZ /\ 1 e. ZZ ) /\ ( 2 e. ZZ /\ 0 e. ZZ ) ) -> ( ( ( 0 =/= 2 /\ 0 =/= 0 ) \/ ( 1 =/= 2 /\ 1 =/= 0 ) ) -> { 0 , 1 } =/= { 2 , 0 } ) )
21	16, 19, 20	mp2 16	. . 3 |- { 0 , 1 } =/= { 2 , 0 }
22		3nn 9289	. . . . . 6 |- 3 e. NN
23	1, 22	pm3.2i 272	. . . . 5 |- ( 0 e. ZZ /\ 3 e. NN )
24	3, 23	pm3.2i 272	. . . 4 |- ( ( 0 e. ZZ /\ 1 e. ZZ ) /\ ( 0 e. ZZ /\ 3 e. NN ) )
25		1re 8161	. . . . . . 7 |- 1 e. RR
26		1lt3 9298	. . . . . . 7 |- 1 < 3
27	25, 26	ltneii 8259	. . . . . 6 |- 1 =/= 3
28	7, 27	pm3.2i 272	. . . . 5 |- ( 1 =/= 0 /\ 1 =/= 3 )
29	28	olci 737	. . . 4 |- ( ( 0 =/= 0 /\ 0 =/= 3 ) \/ ( 1 =/= 0 /\ 1 =/= 3 ) )
30		prneimg 3852	. . . 4 |- ( ( ( 0 e. ZZ /\ 1 e. ZZ ) /\ ( 0 e. ZZ /\ 3 e. NN ) ) -> ( ( ( 0 =/= 0 /\ 0 =/= 3 ) \/ ( 1 =/= 0 /\ 1 =/= 3 ) ) -> { 0 , 1 } =/= { 0 , 3 } ) )
31	24, 29, 30	mp2 16	. . 3 |- { 0 , 1 } =/= { 0 , 3 }
32	14, 21, 31	3pm3.2i 1199	. 2 |- ( { 0 , 1 } =/= { 1 , 2 } /\ { 0 , 1 } =/= { 2 , 0 } /\ { 0 , 1 } =/= { 0 , 3 } )
33	5, 15	pm3.2i 272	. . . 4 |- ( ( 1 e. ZZ /\ 2 e. ZZ ) /\ ( 2 e. ZZ /\ 0 e. ZZ ) )
34	18	orci 736	. . . 4 |- ( ( 1 =/= 2 /\ 1 =/= 0 ) \/ ( 2 =/= 2 /\ 2 =/= 0 ) )
35		prneimg 3852	. . . 4 |- ( ( ( 1 e. ZZ /\ 2 e. ZZ ) /\ ( 2 e. ZZ /\ 0 e. ZZ ) ) -> ( ( ( 1 =/= 2 /\ 1 =/= 0 ) \/ ( 2 =/= 2 /\ 2 =/= 0 ) ) -> { 1 , 2 } =/= { 2 , 0 } ) )
36	33, 34, 35	mp2 16	. . 3 |- { 1 , 2 } =/= { 2 , 0 }
37	5, 23	pm3.2i 272	. . . 4 |- ( ( 1 e. ZZ /\ 2 e. ZZ ) /\ ( 0 e. ZZ /\ 3 e. NN ) )
38	28	orci 736	. . . 4 |- ( ( 1 =/= 0 /\ 1 =/= 3 ) \/ ( 2 =/= 0 /\ 2 =/= 3 ) )
39		prneimg 3852	. . . 4 |- ( ( ( 1 e. ZZ /\ 2 e. ZZ ) /\ ( 0 e. ZZ /\ 3 e. NN ) ) -> ( ( ( 1 =/= 0 /\ 1 =/= 3 ) \/ ( 2 =/= 0 /\ 2 =/= 3 ) ) -> { 1 , 2 } =/= { 0 , 3 } ) )
40	37, 38, 39	mp2 16	. . 3 |- { 1 , 2 } =/= { 0 , 3 }
41	15, 23	pm3.2i 272	. . . 4 |- ( ( 2 e. ZZ /\ 0 e. ZZ ) /\ ( 0 e. ZZ /\ 3 e. NN ) )
42		2re 9196	. . . . . . 7 |- 2 e. RR
43		2lt3 9297	. . . . . . 7 |- 2 < 3
44	42, 43	ltneii 8259	. . . . . 6 |- 2 =/= 3
45	9, 44	pm3.2i 272	. . . . 5 |- ( 2 =/= 0 /\ 2 =/= 3 )
46	45	orci 736	. . . 4 |- ( ( 2 =/= 0 /\ 2 =/= 3 ) \/ ( 0 =/= 0 /\ 0 =/= 3 ) )
47		prneimg 3852	. . . 4 |- ( ( ( 2 e. ZZ /\ 0 e. ZZ ) /\ ( 0 e. ZZ /\ 3 e. NN ) ) -> ( ( ( 2 =/= 0 /\ 2 =/= 3 ) \/ ( 0 =/= 0 /\ 0 =/= 3 ) ) -> { 2 , 0 } =/= { 0 , 3 } ) )
48	41, 46, 47	mp2 16	. . 3 |- { 2 , 0 } =/= { 0 , 3 }
49	36, 40, 48	3pm3.2i 1199	. 2 |- ( { 1 , 2 } =/= { 2 , 0 } /\ { 1 , 2 } =/= { 0 , 3 } /\ { 2 , 0 } =/= { 0 , 3 } )
50	32, 49	pm3.2i 272	1 |- ( ( { 0 , 1 } =/= { 1 , 2 } /\ { 0 , 1 } =/= { 2 , 0 } /\ { 0 , 1 } =/= { 0 , 3 } ) /\ ( { 1 , 2 } =/= { 2 , 0 } /\ { 1 , 2 } =/= { 0 , 3 } /\ { 2 , 0 } =/= { 0 , 3 } ) )

Syntax hints:   /\ wa 104   \/ wo 713   /\ w3a 1002   e. wcel 2200   =/= wne 2400   {cpr 3667   0cc0 8015   1c1 8016   NNcn 9126   2c2 9177   3c3 9178   ZZcz 9462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-iota 5281  df-fun 5323  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-2 9185  df-3 9186  df-z 9463

This theorem is referenced by: (None)