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Theorem usgraexmpldifpr 24817
Description: Lemma for usgraexmpl 24818: 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 10916 . . . . . 6  |-  0  e.  ZZ
2 1z 10935 . . . . . 6  |-  1  e.  ZZ
31, 2pm3.2i 453 . . . . 5  |-  ( 0  e.  ZZ  /\  1  e.  ZZ )
4 2z 10937 . . . . . 6  |-  2  e.  ZZ
52, 4pm3.2i 453 . . . . 5  |-  ( 1  e.  ZZ  /\  2  e.  ZZ )
63, 5pm3.2i 453 . . . 4  |-  ( ( 0  e.  ZZ  /\  1  e.  ZZ )  /\  ( 1  e.  ZZ  /\  2  e.  ZZ ) )
7 ax-1ne0 9591 . . . . . . 7  |-  1  =/=  0
87necomi 2673 . . . . . 6  |-  0  =/=  1
9 2ne0 10669 . . . . . . 7  |-  2  =/=  0
109necomi 2673 . . . . . 6  |-  0  =/=  2
118, 10pm3.2i 453 . . . . 5  |-  ( 0  =/=  1  /\  0  =/=  2 )
1211orci 388 . . . 4  |-  ( ( 0  =/=  1  /\  0  =/=  2 )  \/  ( 1  =/=  1  /\  1  =/=  2 ) )
13 prneimg 4153 . . . 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 453 . . . . 5  |-  ( 2  e.  ZZ  /\  0  e.  ZZ )
163, 15pm3.2i 453 . . . 4  |-  ( ( 0  e.  ZZ  /\  1  e.  ZZ )  /\  ( 2  e.  ZZ  /\  0  e.  ZZ ) )
17 1ne2 10789 . . . . . 6  |-  1  =/=  2
1817, 7pm3.2i 453 . . . . 5  |-  ( 1  =/=  2  /\  1  =/=  0 )
1918olci 389 . . . 4  |-  ( ( 0  =/=  2  /\  0  =/=  0 )  \/  ( 1  =/=  2  /\  1  =/=  0 ) )
20 prneimg 4153 . . . 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 10735 . . . . . 6  |-  3  e.  NN
231, 22pm3.2i 453 . . . . 5  |-  ( 0  e.  ZZ  /\  3  e.  NN )
243, 23pm3.2i 453 . . . 4  |-  ( ( 0  e.  ZZ  /\  1  e.  ZZ )  /\  ( 0  e.  ZZ  /\  3  e.  NN ) )
25 1re 9625 . . . . . . 7  |-  1  e.  RR
26 1lt3 10745 . . . . . . 7  |-  1  <  3
2725, 26ltneii 9729 . . . . . 6  |-  1  =/=  3
287, 27pm3.2i 453 . . . . 5  |-  ( 1  =/=  0  /\  1  =/=  3 )
2928olci 389 . . . 4  |-  ( ( 0  =/=  0  /\  0  =/=  3 )  \/  ( 1  =/=  0  /\  1  =/=  3 ) )
30 prneimg 4153 . . . 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 1175 . 2  |-  ( { 0 ,  1 }  =/=  { 1 ,  2 }  /\  {
0 ,  1 }  =/=  { 2 ,  0 }  /\  {
0 ,  1 }  =/=  { 0 ,  3 } )
335, 15pm3.2i 453 . . . 4  |-  ( ( 1  e.  ZZ  /\  2  e.  ZZ )  /\  ( 2  e.  ZZ  /\  0  e.  ZZ ) )
3418orci 388 . . . 4  |-  ( ( 1  =/=  2  /\  1  =/=  0 )  \/  ( 2  =/=  2  /\  2  =/=  0 ) )
35 prneimg 4153 . . . 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 453 . . . 4  |-  ( ( 1  e.  ZZ  /\  2  e.  ZZ )  /\  ( 0  e.  ZZ  /\  3  e.  NN ) )
3828orci 388 . . . 4  |-  ( ( 1  =/=  0  /\  1  =/=  3 )  \/  ( 2  =/=  0  /\  2  =/=  3 ) )
39 prneimg 4153 . . . 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 453 . . . 4  |-  ( ( 2  e.  ZZ  /\  0  e.  ZZ )  /\  ( 0  e.  ZZ  /\  3  e.  NN ) )
42 2re 10646 . . . . . . 7  |-  2  e.  RR
43 2lt3 10744 . . . . . . 7  |-  2  <  3
4442, 43ltneii 9729 . . . . . 6  |-  2  =/=  3
459, 44pm3.2i 453 . . . . 5  |-  ( 2  =/=  0  /\  2  =/=  3 )
4645orci 388 . . . 4  |-  ( ( 2  =/=  0  /\  2  =/=  3 )  \/  ( 0  =/=  0  /\  0  =/=  3 ) )
47 prneimg 4153 . . . 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 1175 . 2  |-  ( { 1 ,  2 }  =/=  { 2 ,  0 }  /\  {
1 ,  2 }  =/=  { 0 ,  3 }  /\  {
2 ,  0 }  =/=  { 0 ,  3 } )
5032, 49pm3.2i 453 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 366    /\ wa 367    /\ w3a 974    e. wcel 1842    =/= wne 2598   {cpr 3974   0cc0 9522   1c1 9523   NNcn 10576   2c2 10626   3c3 10627   ZZcz 10905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-z 10906
This theorem is referenced by:  usgraexmpl  24818  usgraexmpledg  24820
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