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Theorem constr3lem6 21628
Description: Lemma for constr3pthlem3 21636. (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 21626 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( ( P `
 0 )  =  A  /\  ( P `
 1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) ) )
4 id 20 . . . . . . . 8  |-  ( A  =/=  B  ->  A  =/=  B )
54ancli 535 . . . . . . 7  |-  ( A  =/=  B  ->  ( A  =/=  B  /\  A  =/=  B ) )
6 necom 2679 . . . . . . . 8  |-  ( C  =/=  A  <->  A  =/=  C )
7 id 20 . . . . . . . . 9  |-  ( A  =/=  C  ->  A  =/=  C )
87ancli 535 . . . . . . . 8  |-  ( A  =/=  C  ->  ( A  =/=  C  /\  A  =/=  C ) )
96, 8sylbi 188 . . . . . . 7  |-  ( C  =/=  A  ->  ( A  =/=  C  /\  A  =/=  C ) )
105, 9anim12i 550 . . . . . 6  |-  ( ( A  =/=  B  /\  C  =/=  A )  -> 
( ( A  =/= 
B  /\  A  =/=  B )  /\  ( A  =/=  C  /\  A  =/=  C ) ) )
11103adant2 976 . . . . 5  |-  ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A )  ->  (
( A  =/=  B  /\  A  =/=  B
)  /\  ( A  =/=  C  /\  A  =/= 
C ) ) )
12 simpl 444 . . . . . . . . 9  |-  ( ( ( P `  0
)  =  A  /\  ( P `  1 )  =  B )  -> 
( P `  0
)  =  A )
13 simpr 448 . . . . . . . . 9  |-  ( ( ( P `  0
)  =  A  /\  ( P `  1 )  =  B )  -> 
( P `  1
)  =  B )
1412, 13neeq12d 2613 . . . . . . . 8  |-  ( ( ( P `  0
)  =  A  /\  ( P `  1 )  =  B )  -> 
( ( P ` 
0 )  =/=  ( P `  1 )  <->  A  =/=  B ) )
1514adantr 452 . . . . . . 7  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( P ` 
0 )  =/=  ( P `  1 )  <->  A  =/=  B ) )
16 simpr 448 . . . . . . . . 9  |-  ( ( ( P `  2
)  =  C  /\  ( P `  3 )  =  A )  -> 
( P `  3
)  =  A )
1716adantl 453 . . . . . . . 8  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  3
)  =  A )
1813adantr 452 . . . . . . . 8  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  1
)  =  B )
1917, 18neeq12d 2613 . . . . . . 7  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( P ` 
3 )  =/=  ( P `  1 )  <->  A  =/=  B ) )
2015, 19anbi12d 692 . . . . . 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 452 . . . . . . . 8  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  0
)  =  A )
22 simpl 444 . . . . . . . . 9  |-  ( ( ( P `  2
)  =  C  /\  ( P `  3 )  =  A )  -> 
( P `  2
)  =  C )
2322adantl 453 . . . . . . . 8  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  2
)  =  C )
2421, 23neeq12d 2613 . . . . . . 7  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( P ` 
0 )  =/=  ( P `  2 )  <->  A  =/=  C ) )
2516, 22neeq12d 2613 . . . . . . . 8  |-  ( ( ( P `  2
)  =  C  /\  ( P `  3 )  =  A )  -> 
( ( P ` 
3 )  =/=  ( P `  2 )  <->  A  =/=  C ) )
2625adantl 453 . . . . . . 7  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( P ` 
3 )  =/=  ( P `  2 )  <->  A  =/=  C ) )
2724, 26anbi12d 692 . . . . . 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 692 . . . . 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 213 . . . 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 16 . . 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 419 . 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 3862 . 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 16 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    u. cun 3310    i^i cin 3311   (/)c0 3620   {cpr 3807   {ctp 3808   <.cop 3809   `'ccnv 4869   ` cfv 5446   0cc0 8982   1c1 8983   2c2 10041   3c3 10042
This theorem is referenced by:  constr3pthlem3  21636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275
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