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Theorem usgraexmpldifpr 24076
Description: Lemma for usgraexmpl 24077: all "edges" are different. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
Assertion
Ref Expression
usgraexmpldifpr  |-  ( ( { 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 usgraexmpldifpr
StepHypRef Expression
1 0z 10871 . . . . . 6  |-  0  e.  ZZ
2 1z 10890 . . . . . 6  |-  1  e.  ZZ
31, 2pm3.2i 455 . . . . 5  |-  ( 0  e.  ZZ  /\  1  e.  ZZ )
4 2z 10892 . . . . . 6  |-  2  e.  ZZ
52, 4pm3.2i 455 . . . . 5  |-  ( 1  e.  ZZ  /\  2  e.  ZZ )
63, 5pm3.2i 455 . . . 4  |-  ( ( 0  e.  ZZ  /\  1  e.  ZZ )  /\  ( 1  e.  ZZ  /\  2  e.  ZZ ) )
7 ax-1ne0 9557 . . . . . . 7  |-  1  =/=  0
87necomi 2737 . . . . . 6  |-  0  =/=  1
9 2ne0 10624 . . . . . . 7  |-  2  =/=  0
109necomi 2737 . . . . . 6  |-  0  =/=  2
118, 10pm3.2i 455 . . . . 5  |-  ( 0  =/=  1  /\  0  =/=  2 )
1211orci 390 . . . 4  |-  ( ( 0  =/=  1  /\  0  =/=  2 )  \/  ( 1  =/=  1  /\  1  =/=  2 ) )
13 prneimg 4207 . . . 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 } ) )
146, 12, 13mp2 9 . . 3  |-  { 0 ,  1 }  =/=  { 1 ,  2 }
154, 1pm3.2i 455 . . . . 5  |-  ( 2  e.  ZZ  /\  0  e.  ZZ )
163, 15pm3.2i 455 . . . 4  |-  ( ( 0  e.  ZZ  /\  1  e.  ZZ )  /\  ( 2  e.  ZZ  /\  0  e.  ZZ ) )
17 1ne2 10744 . . . . . 6  |-  1  =/=  2
1817, 7pm3.2i 455 . . . . 5  |-  ( 1  =/=  2  /\  1  =/=  0 )
1918olci 391 . . . 4  |-  ( ( 0  =/=  2  /\  0  =/=  0 )  \/  ( 1  =/=  2  /\  1  =/=  0 ) )
20 prneimg 4207 . . . 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 } ) )
2116, 19, 20mp2 9 . . 3  |-  { 0 ,  1 }  =/=  { 2 ,  0 }
22 3nn 10690 . . . . . 6  |-  3  e.  NN
231, 22pm3.2i 455 . . . . 5  |-  ( 0  e.  ZZ  /\  3  e.  NN )
243, 23pm3.2i 455 . . . 4  |-  ( ( 0  e.  ZZ  /\  1  e.  ZZ )  /\  ( 0  e.  ZZ  /\  3  e.  NN ) )
25 1re 9591 . . . . . . 7  |-  1  e.  RR
26 1lt3 10700 . . . . . . 7  |-  1  <  3
2725, 26ltneii 9693 . . . . . 6  |-  1  =/=  3
287, 27pm3.2i 455 . . . . 5  |-  ( 1  =/=  0  /\  1  =/=  3 )
2928olci 391 . . . 4  |-  ( ( 0  =/=  0  /\  0  =/=  3 )  \/  ( 1  =/=  0  /\  1  =/=  3 ) )
30 prneimg 4207 . . . 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 } ) )
3124, 29, 30mp2 9 . . 3  |-  { 0 ,  1 }  =/=  { 0 ,  3 }
3214, 21, 313pm3.2i 1174 . 2  |-  ( { 0 ,  1 }  =/=  { 1 ,  2 }  /\  {
0 ,  1 }  =/=  { 2 ,  0 }  /\  {
0 ,  1 }  =/=  { 0 ,  3 } )
335, 15pm3.2i 455 . . . 4  |-  ( ( 1  e.  ZZ  /\  2  e.  ZZ )  /\  ( 2  e.  ZZ  /\  0  e.  ZZ ) )
3418orci 390 . . . 4  |-  ( ( 1  =/=  2  /\  1  =/=  0 )  \/  ( 2  =/=  2  /\  2  =/=  0 ) )
35 prneimg 4207 . . . 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 } ) )
3633, 34, 35mp2 9 . . 3  |-  { 1 ,  2 }  =/=  { 2 ,  0 }
375, 23pm3.2i 455 . . . 4  |-  ( ( 1  e.  ZZ  /\  2  e.  ZZ )  /\  ( 0  e.  ZZ  /\  3  e.  NN ) )
3828orci 390 . . . 4  |-  ( ( 1  =/=  0  /\  1  =/=  3 )  \/  ( 2  =/=  0  /\  2  =/=  3 ) )
39 prneimg 4207 . . . 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 } ) )
4037, 38, 39mp2 9 . . 3  |-  { 1 ,  2 }  =/=  { 0 ,  3 }
4115, 23pm3.2i 455 . . . 4  |-  ( ( 2  e.  ZZ  /\  0  e.  ZZ )  /\  ( 0  e.  ZZ  /\  3  e.  NN ) )
42 2re 10601 . . . . . . 7  |-  2  e.  RR
43 2lt3 10699 . . . . . . 7  |-  2  <  3
4442, 43ltneii 9693 . . . . . 6  |-  2  =/=  3
459, 44pm3.2i 455 . . . . 5  |-  ( 2  =/=  0  /\  2  =/=  3 )
4645orci 390 . . . 4  |-  ( ( 2  =/=  0  /\  2  =/=  3 )  \/  ( 0  =/=  0  /\  0  =/=  3 ) )
47 prneimg 4207 . . . 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 } ) )
4841, 46, 47mp2 9 . . 3  |-  { 2 ,  0 }  =/=  { 0 ,  3 }
4936, 40, 483pm3.2i 1174 . 2  |-  ( { 1 ,  2 }  =/=  { 2 ,  0 }  /\  {
1 ,  2 }  =/=  { 0 ,  3 }  /\  {
2 ,  0 }  =/=  { 0 ,  3 } )
5032, 49pm3.2i 455 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 } ) )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    /\ w3a 973    e. wcel 1767    =/= wne 2662   {cpr 4029   0cc0 9488   1c1 9489   NNcn 10532   2c2 10581   3c3 10582   ZZcz 10860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-z 10861
This theorem is referenced by:  usgraexmpl  24077  usgraexmpledg  24079
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