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Theorem usgraexmpldifpr 23453
Description: Lemma for usgraexmpl 23454: 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 10758 . . . . . 6  |-  0  e.  ZZ
2 1z 10777 . . . . . 6  |-  1  e.  ZZ
31, 2pm3.2i 455 . . . . 5  |-  ( 0  e.  ZZ  /\  1  e.  ZZ )
4 2z 10779 . . . . . 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 9452 . . . . . . 7  |-  1  =/=  0
87necomi 2718 . . . . . 6  |-  0  =/=  1
9 2ne0 10515 . . . . . . 7  |-  2  =/=  0
109necomi 2718 . . . . . 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 4151 . . . 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 10635 . . . . . 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 4151 . . . 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 10581 . . . . . 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 9486 . . . . . . 7  |-  1  e.  RR
26 1lt3 10591 . . . . . . 7  |-  1  <  3
2725, 26ltneii 9588 . . . . . 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 4151 . . . 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 1166 . 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 4151 . . . 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 4151 . . . 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 10492 . . . . . . 7  |-  2  e.  RR
43 2lt3 10590 . . . . . . 7  |-  2  <  3
4442, 43ltneii 9588 . . . . . 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 4151 . . . 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 1166 . 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 965    e. wcel 1758    =/= wne 2644   {cpr 3977   0cc0 9383   1c1 9384   NNcn 10423   2c2 10472   3c3 10473   ZZcz 10747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-z 10748
This theorem is referenced by:  usgraexmpl  23454
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