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Theorem isconcl5ab 25470
Description: Two distinct lines intersect in at most one point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
isconcl5a.1  |-  L  =  (PLines `  F )
isconcl5a.2  |-  P  =  (PPoints `  F )
isconcl5a.3  |-  ( ph  ->  F  e. Ig )
isconcl5a.4  |-  ( ph  -> 
L1  e.  L )
isconcl5a.5  |-  ( ph  ->  L 2  e.  L
)
isconcl5a.6  |-  ( ph  -> 
L1  =/=  L 2
)
Assertion
Ref Expression
isconcl5ab  |-  ( ph  ->  E* p ( p  e.  L1  /\  p  e.  L 2 ) )
Distinct variable groups:    p, L1    p, L 2    ph, p
Allowed substitution hints:    P( p)    F( p)    L( p)

Proof of Theorem isconcl5ab
StepHypRef Expression
1 vex 2766 . . . . . . 7  |-  q  e. 
_V
2 eleq1 2318 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  L1  <->  q  e.  L1 ) )
3 eleq1 2318 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  L 2  <->  q  e.  L 2 )
)
42, 3anbi12d 694 . . . . . . 7  |-  ( p  =  q  ->  (
( p  e.  L1  /\  p  e.  L 2
)  <->  ( q  e.  L1  /\  q  e.  L 2 ) ) )
51, 4sbcie 3000 . . . . . 6  |-  ( [. q  /  p ]. (
p  e.  L1  /\  p  e.  L 2 )  <->  ( q  e.  L1  /\  q  e.  L 2 ) )
6 idd 23 . . . . . . . . 9  |-  ( ( p  e.  L1  /\  p  e.  L 2 )  -> 
( p  =  q  ->  p  =  q ) )
76a1ii 26 . . . . . . . 8  |-  ( ph  ->  ( ( q  e.  L1  /\  q  e.  L 2 )  ->  (
( p  e.  L1  /\  p  e.  L 2
)  ->  ( p  =  q  ->  p  =  q ) ) ) )
87com14 84 . . . . . . 7  |-  ( p  =  q  ->  (
( q  e.  L1  /\  q  e.  L 2
)  ->  ( (
p  e.  L1  /\  p  e.  L 2 )  -> 
( ph  ->  p  =  q ) ) ) )
9 isconcl5a.2 . . . . . . . . . . . . 13  |-  P  =  (PPoints `  F )
10 isconcl5a.1 . . . . . . . . . . . . 13  |-  L  =  (PLines `  F )
11 eqid 2258 . . . . . . . . . . . . 13  |-  ( line `  F )  =  (
line `  F )
12 isconcl5a.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e. Ig )
1312adantl 454 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  F  e. Ig )
14 isconcl5a.4 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  -> 
L1  e.  L )
159, 10, 12, 14isig12 25432 . . . . . . . . . . . . . . . . . 18  |-  ( ph  -> 
L1  C_  P )
1615sseld 3154 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( p  e.  L1  ->  p  e.  P ) )
1716com12 29 . . . . . . . . . . . . . . . 16  |-  ( p  e.  L1  ->  ( ph  ->  p  e.  P ) )
1817adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  L1  /\  p  e.  L 2 )  -> 
( ph  ->  p  e.  P ) )
19183ad2ant3 983 . . . . . . . . . . . . . 14  |-  ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  ->  ( ph  ->  p  e.  P ) )
2019imp 420 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  p  e.  P )
2115sseld 3154 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( q  e.  L1  ->  q  e.  P ) )
2221com12 29 . . . . . . . . . . . . . . . 16  |-  ( q  e.  L1  ->  ( ph  ->  q  e.  P ) )
2322adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  L1  /\  q  e.  L 2 )  -> 
( ph  ->  q  e.  P ) )
24233ad2ant2 982 . . . . . . . . . . . . . 14  |-  ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  ->  ( ph  ->  q  e.  P ) )
2524imp 420 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  -> 
q  e.  P )
26 simpl1 963 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  p  =/=  q )
27 simpl3l 1015 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  p  e.  L1 )
28 simpl2l 1013 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  -> 
q  e.  L1 )
2914adantl 454 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L1  e.  L )
309, 10, 11, 13, 20, 25, 26, 27, 28, 29lineval42 25448 . . . . . . . . . . . 12  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L1  =  ( p (
line `  F )
q ) )
31 simpl3r 1016 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  p  e.  L 2
)
32 simpl2r 1014 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  -> 
q  e.  L 2
)
33 isconcl5a.5 . . . . . . . . . . . . . 14  |-  ( ph  ->  L 2  e.  L
)
3433adantl 454 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L 2  e.  L
)
359, 10, 11, 13, 20, 25, 26, 31, 32, 34lineval42 25448 . . . . . . . . . . . 12  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L 2  =  (
p ( line `  F
) q ) )
3630, 35eqtr4d 2293 . . . . . . . . . . 11  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L1  =  L 2 )
37 isconcl5a.6 . . . . . . . . . . . . 13  |-  ( ph  -> 
L1  =/=  L 2
)
3837neneqd 2437 . . . . . . . . . . . 12  |-  ( ph  ->  -.  L1  =  L 2 )
3938adantl 454 . . . . . . . . . . 11  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  -.  L1  =  L 2
)
4036, 39jca 520 . . . . . . . . . 10  |-  ( ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  -> 
( L1  =  L 2  /\  -.  L1  =  L 2 ) )
4140ex 425 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  ->  ( ph  ->  (
L1  =  L 2  /\  -.  L1  =  L 2 ) ) )
42 pm2.24 103 . . . . . . . . . 10  |-  ( L1  =  L 2  ->  ( -.  L1  =  L 2  ->  p  =  q ) )
4342imp 420 . . . . . . . . 9  |-  ( (
L1  =  L 2  /\  -.  L1  =  L 2 )  ->  p  =  q )
4441, 43syl6 31 . . . . . . . 8  |-  ( ( p  =/=  q  /\  ( q  e.  L1  /\  q  e.  L 2
)  /\  ( p  e.  L1  /\  p  e.  L 2 ) )  ->  ( ph  ->  p  =  q ) )
45443exp 1155 . . . . . . 7  |-  ( p  =/=  q  ->  (
( q  e.  L1  /\  q  e.  L 2
)  ->  ( (
p  e.  L1  /\  p  e.  L 2 )  -> 
( ph  ->  p  =  q ) ) ) )
468, 45pm2.61ine 2497 . . . . . 6  |-  ( ( q  e.  L1  /\  q  e.  L 2 )  -> 
( ( p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  p  =  q ) ) )
475, 46sylbi 189 . . . . 5  |-  ( [. q  /  p ]. (
p  e.  L1  /\  p  e.  L 2 )  -> 
( ( p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  p  =  q ) ) )
4847impcom 421 . . . 4  |-  ( ( ( p  e.  L1  /\  p  e.  L 2
)  /\  [. q  /  p ]. ( p  e.  L1  /\  p  e.  L 2 ) )  -> 
( ph  ->  p  =  q ) )
4948com12 29 . . 3  |-  ( ph  ->  ( ( ( p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
5049alrimivv 2014 . 2  |-  ( ph  ->  A. p A. q
( ( ( p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
51 nfv 1629 . . . 4  |-  F/ q ( p  e.  L1  /\  p  e.  L 2
)
5251mo3 2149 . . 3  |-  ( E* p ( p  e.  L1  /\  p  e.  L 2 )  <->  A. p A. q ( ( ( p  e.  L1  /\  p  e.  L 2 )  /\  [ q  /  p ]
( p  e.  L1  /\  p  e.  L 2
) )  ->  p  =  q ) )
53 sbsbc 2970 . . . . . 6  |-  ( [ q  /  p ]
( p  e.  L1  /\  p  e.  L 2
)  <->  [. q  /  p ]. ( p  e.  L1  /\  p  e.  L 2
) )
5453anbi2i 678 . . . . 5  |-  ( ( ( p  e.  L1  /\  p  e.  L 2
)  /\  [ q  /  p ] ( p  e.  L1  /\  p  e.  L 2 ) )  <-> 
( ( p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  L1  /\  p  e.  L 2 ) ) )
5554imbi1i 317 . . . 4  |-  ( ( ( ( p  e.  L1  /\  p  e.  L 2 )  /\  [
q  /  p ]
( p  e.  L1  /\  p  e.  L 2
) )  ->  p  =  q )  <->  ( (
( p  e.  L1  /\  p  e.  L 2
)  /\  [. q  /  p ]. ( p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q )
)
56552albii 1555 . . 3  |-  ( A. p A. q ( ( ( p  e.  L1  /\  p  e.  L 2
)  /\  [ q  /  p ] ( p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q )  <->  A. p A. q
( ( ( p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
5752, 56bitri 242 . 2  |-  ( E* p ( p  e.  L1  /\  p  e.  L 2 )  <->  A. p A. q ( ( ( p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
5850, 57sylibr 205 1  |-  ( ph  ->  E* p ( p  e.  L1  /\  p  e.  L 2 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939   A.wal 1532    = wceq 1619    e. wcel 1621   [wsb 1883   E*wmo 2119    =/= wne 2421   [.wsbc 2966   ` cfv 4673  (class class class)co 5792  PPointscpoints 25424  PLinescplines 25426  Igcig 25428   linecline 25444
This theorem is referenced by:  isconcl6ab  25472
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-ig2 25429  df-li 25445
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