Theorem usgrexmpldifpr 16103

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 9490	. . . . . 6 |- 0 e. ZZ
2		1z 9505	. . . . . 6 |- 1 e. ZZ
3	1, 2	pm3.2i 272	. . . . 5 |- ( 0 e. ZZ /\ 1 e. ZZ )
4		2z 9507	. . . . . 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 9211	. . . . . . 7 |- 1 =/= 0
8	7	necomi 2487	. . . . . 6 |- 0 =/= 1
9		2ne0 9235	. . . . . . 7 |- 2 =/= 0
10	9	necomi 2487	. . . . . 6 |- 0 =/= 2
11	8, 10	pm3.2i 272	. . . . 5 |- ( 0 =/= 1 /\ 0 =/= 2 )
12	11	orci 738	. . . 4 |- ( ( 0 =/= 1 /\ 0 =/= 2 ) \/ ( 1 =/= 1 /\ 1 =/= 2 ) )
13		prneimg 3857	. . . 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 9350	. . . . . 6 |- 1 =/= 2
18	17, 7	pm3.2i 272	. . . . 5 |- ( 1 =/= 2 /\ 1 =/= 0 )
19	18	olci 739	. . . 4 |- ( ( 0 =/= 2 /\ 0 =/= 0 ) \/ ( 1 =/= 2 /\ 1 =/= 0 ) )
20		prneimg 3857	. . . 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 9306	. . . . . 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 8178	. . . . . . 7 |- 1 e. RR
26		1lt3 9315	. . . . . . 7 |- 1 < 3
27	25, 26	ltneii 8276	. . . . . 6 |- 1 =/= 3
28	7, 27	pm3.2i 272	. . . . 5 |- ( 1 =/= 0 /\ 1 =/= 3 )
29	28	olci 739	. . . 4 |- ( ( 0 =/= 0 /\ 0 =/= 3 ) \/ ( 1 =/= 0 /\ 1 =/= 3 ) )
30		prneimg 3857	. . . 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 1201	. 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 738	. . . 4 |- ( ( 1 =/= 2 /\ 1 =/= 0 ) \/ ( 2 =/= 2 /\ 2 =/= 0 ) )
35		prneimg 3857	. . . 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 738	. . . 4 |- ( ( 1 =/= 0 /\ 1 =/= 3 ) \/ ( 2 =/= 0 /\ 2 =/= 3 ) )
39		prneimg 3857	. . . 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 9213	. . . . . . 7 |- 2 e. RR
43		2lt3 9314	. . . . . . 7 |- 2 < 3
44	42, 43	ltneii 8276	. . . . . 6 |- 2 =/= 3
45	9, 44	pm3.2i 272	. . . . 5 |- ( 2 =/= 0 /\ 2 =/= 3 )
46	45	orci 738	. . . 4 |- ( ( 2 =/= 0 /\ 2 =/= 3 ) \/ ( 0 =/= 0 /\ 0 =/= 3 ) )
47		prneimg 3857	. . . 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 1201	. 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 715   /\ w3a 1004   e. wcel 2202   =/= wne 2402   {cpr 3670   0cc0 8032   1c1 8033   NNcn 9143   2c2 9194   3c3 9195   ZZcz 9479

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-z 9480

This theorem is referenced by: (None)