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Theorem constr3lem6 25066
Description: Lemma for constr3pthlem3 25074. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
Hypotheses
Ref Expression
constr3cycl.f  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. ,  <. 2 ,  ( `' E `  { C ,  A } ) >. }
constr3cycl.p  |-  P  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  A >. } )
Assertion
Ref Expression
constr3lem6  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C  /\  C  =/=  A ) )  ->  ( { ( P ` 
0 ) ,  ( P `  3 ) }  i^i  { ( P `  1 ) ,  ( P ` 
2 ) } )  =  (/) )

Proof of Theorem constr3lem6
StepHypRef Expression
1 constr3cycl.f . . . . 5  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. ,  <. 2 ,  ( `' E `  { C ,  A } ) >. }
2 constr3cycl.p . . . . 5  |-  P  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  A >. } )
31, 2constr3lem4 25064 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( ( P `
 0 )  =  A  /\  ( P `
 1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) ) )
4 id 22 . . . . . . . 8  |-  ( A  =/=  B  ->  A  =/=  B )
54ancli 549 . . . . . . 7  |-  ( A  =/=  B  ->  ( A  =/=  B  /\  A  =/=  B ) )
6 necom 2672 . . . . . . . 8  |-  ( C  =/=  A  <->  A  =/=  C )
7 id 22 . . . . . . . . 9  |-  ( A  =/=  C  ->  A  =/=  C )
87ancli 549 . . . . . . . 8  |-  ( A  =/=  C  ->  ( A  =/=  C  /\  A  =/=  C ) )
96, 8sylbi 195 . . . . . . 7  |-  ( C  =/=  A  ->  ( A  =/=  C  /\  A  =/=  C ) )
105, 9anim12i 564 . . . . . 6  |-  ( ( A  =/=  B  /\  C  =/=  A )  -> 
( ( A  =/= 
B  /\  A  =/=  B )  /\  ( A  =/=  C  /\  A  =/=  C ) ) )
11103adant2 1016 . . . . 5  |-  ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A )  ->  (
( A  =/=  B  /\  A  =/=  B
)  /\  ( A  =/=  C  /\  A  =/= 
C ) ) )
12 simpl 455 . . . . . . . . 9  |-  ( ( ( P `  0
)  =  A  /\  ( P `  1 )  =  B )  -> 
( P `  0
)  =  A )
13 simpr 459 . . . . . . . . 9  |-  ( ( ( P `  0
)  =  A  /\  ( P `  1 )  =  B )  -> 
( P `  1
)  =  B )
1412, 13neeq12d 2682 . . . . . . . 8  |-  ( ( ( P `  0
)  =  A  /\  ( P `  1 )  =  B )  -> 
( ( P ` 
0 )  =/=  ( P `  1 )  <->  A  =/=  B ) )
1514adantr 463 . . . . . . 7  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( P ` 
0 )  =/=  ( P `  1 )  <->  A  =/=  B ) )
16 simpr 459 . . . . . . . . 9  |-  ( ( ( P `  2
)  =  C  /\  ( P `  3 )  =  A )  -> 
( P `  3
)  =  A )
1716adantl 464 . . . . . . . 8  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  3
)  =  A )
1813adantr 463 . . . . . . . 8  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  1
)  =  B )
1917, 18neeq12d 2682 . . . . . . 7  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( P ` 
3 )  =/=  ( P `  1 )  <->  A  =/=  B ) )
2015, 19anbi12d 709 . . . . . 6  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  3 )  =/=  ( P `  1
) )  <->  ( A  =/=  B  /\  A  =/= 
B ) ) )
2112adantr 463 . . . . . . . 8  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  0
)  =  A )
22 simpl 455 . . . . . . . . 9  |-  ( ( ( P `  2
)  =  C  /\  ( P `  3 )  =  A )  -> 
( P `  2
)  =  C )
2322adantl 464 . . . . . . . 8  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  2
)  =  C )
2421, 23neeq12d 2682 . . . . . . 7  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( P ` 
0 )  =/=  ( P `  2 )  <->  A  =/=  C ) )
2516, 22neeq12d 2682 . . . . . . . 8  |-  ( ( ( P `  2
)  =  C  /\  ( P `  3 )  =  A )  -> 
( ( P ` 
3 )  =/=  ( P `  2 )  <->  A  =/=  C ) )
2625adantl 464 . . . . . . 7  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( P ` 
3 )  =/=  ( P `  2 )  <->  A  =/=  C ) )
2724, 26anbi12d 709 . . . . . 6  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( ( P `
 0 )  =/=  ( P `  2
)  /\  ( P `  3 )  =/=  ( P `  2
) )  <->  ( A  =/=  C  /\  A  =/= 
C ) ) )
2820, 27anbi12d 709 . . . . 5  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( ( ( P `  0 )  =/=  ( P ` 
1 )  /\  ( P `  3 )  =/=  ( P `  1
) )  /\  (
( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  3 )  =/=  ( P ` 
2 ) ) )  <-> 
( ( A  =/= 
B  /\  A  =/=  B )  /\  ( A  =/=  C  /\  A  =/=  C ) ) ) )
2911, 28syl5ibr 221 . . . 4  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( A  =/= 
B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  3 )  =/=  ( P ` 
1 ) )  /\  ( ( P ` 
0 )  =/=  ( P `  2 )  /\  ( P `  3
)  =/=  ( P `
 2 ) ) ) ) )
303, 29syl 17 . . 3  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( A  =/= 
B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  3 )  =/=  ( P ` 
1 ) )  /\  ( ( P ` 
0 )  =/=  ( P `  2 )  /\  ( P `  3
)  =/=  ( P `
 2 ) ) ) ) )
3130imp 427 . 2  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C  /\  C  =/=  A ) )  ->  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  3
)  =/=  ( P `
 1 ) )  /\  ( ( P `
 0 )  =/=  ( P `  2
)  /\  ( P `  3 )  =/=  ( P `  2
) ) ) )
32 disjpr2 4034 . 2  |-  ( ( ( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  3
)  =/=  ( P `
 1 ) )  /\  ( ( P `
 0 )  =/=  ( P `  2
)  /\  ( P `  3 )  =/=  ( P `  2
) ) )  -> 
( { ( P `
 0 ) ,  ( P `  3
) }  i^i  {
( P `  1
) ,  ( P `
 2 ) } )  =  (/) )
3331, 32syl 17 1  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C  /\  C  =/=  A ) )  ->  ( { ( P ` 
0 ) ,  ( P `  3 ) }  i^i  { ( P `  1 ) ,  ( P ` 
2 ) } )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598    u. cun 3412    i^i cin 3413   (/)c0 3738   {cpr 3974   {ctp 3976   <.cop 3978   `'ccnv 4822   ` cfv 5569   0cc0 9522   1c1 9523   2c2 10626   3c3 10627
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-n0 10837  df-z 10906
This theorem is referenced by:  constr3pthlem3  25074
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