Theorem usgrexmpldifpr 16293

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 9593	. . . . . 6 |- 0 e. ZZ
2		1z 9608	. . . . . 6 |- 1 e. ZZ
3	1, 2	pm3.2i 272	. . . . 5 |- ( 0 e. ZZ /\ 1 e. ZZ )
4		2z 9610	. . . . . 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 9310	. . . . . . 7 |- 1 =/= 0
8	7	necomi 2499	. . . . . 6 |- 0 =/= 1
9		2ne0 9334	. . . . . . 7 |- 2 =/= 0
10	9	necomi 2499	. . . . . 6 |- 0 =/= 2
11	8, 10	pm3.2i 272	. . . . 5 |- ( 0 =/= 1 /\ 0 =/= 2 )
12	11	orci 739	. . . 4 |- ( ( 0 =/= 1 /\ 0 =/= 2 ) \/ ( 1 =/= 1 /\ 1 =/= 2 ) )
13		prneimg 3880	. . . 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 9449	. . . . . 6 |- 1 =/= 2
18	17, 7	pm3.2i 272	. . . . 5 |- ( 1 =/= 2 /\ 1 =/= 0 )
19	18	olci 740	. . . 4 |- ( ( 0 =/= 2 /\ 0 =/= 0 ) \/ ( 1 =/= 2 /\ 1 =/= 0 ) )
20		prneimg 3880	. . . 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 9405	. . . . . 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 8278	. . . . . . 7 |- 1 e. RR
26		1lt3 9414	. . . . . . 7 |- 1 < 3
27	25, 26	ltneii 8375	. . . . . 6 |- 1 =/= 3
28	7, 27	pm3.2i 272	. . . . 5 |- ( 1 =/= 0 /\ 1 =/= 3 )
29	28	olci 740	. . . 4 |- ( ( 0 =/= 0 /\ 0 =/= 3 ) \/ ( 1 =/= 0 /\ 1 =/= 3 ) )
30		prneimg 3880	. . . 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 1202	. 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 739	. . . 4 |- ( ( 1 =/= 2 /\ 1 =/= 0 ) \/ ( 2 =/= 2 /\ 2 =/= 0 ) )
35		prneimg 3880	. . . 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 739	. . . 4 |- ( ( 1 =/= 0 /\ 1 =/= 3 ) \/ ( 2 =/= 0 /\ 2 =/= 3 ) )
39		prneimg 3880	. . . 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 9312	. . . . . . 7 |- 2 e. RR
43		2lt3 9413	. . . . . . 7 |- 2 < 3
44	42, 43	ltneii 8375	. . . . . 6 |- 2 =/= 3
45	9, 44	pm3.2i 272	. . . . 5 |- ( 2 =/= 0 /\ 2 =/= 3 )
46	45	orci 739	. . . 4 |- ( ( 2 =/= 0 /\ 2 =/= 3 ) \/ ( 0 =/= 0 /\ 0 =/= 3 ) )
47		prneimg 3880	. . . 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 1202	. 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 716   /\ w3a 1005   e. wcel 2205   =/= wne 2414   {cpr 3692   0cc0 8132   1c1 8133   NNcn 9242   2c2 9293   3c3 9294   ZZcz 9582

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-ltadd 8248

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-inn 9243  df-2 9301  df-3 9302  df-z 9583

This theorem is referenced by: (None)