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Theorem isconcl5a 26101
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
isconcl5a  |-  ( ph  ->  E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
Distinct variable groups:    p, L1    p, L 2    P, p    ph, p
Allowed substitution hints:    F( p)    L( p)

Proof of Theorem isconcl5a
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . . . 7  |-  q  e. 
_V
2 eleq1 2343 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  P  <->  q  e.  P ) )
3 eleq1 2343 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  L1  <->  q  e.  L1 ) )
4 eleq1 2343 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  L 2  <->  q  e.  L 2 )
)
52, 3, 43anbi123d 1252 . . . . . . 7  |-  ( p  =  q  ->  (
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  <->  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 ) ) )
61, 5sbcie 3025 . . . . . 6  |-  ( [. q  /  p ]. (
p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  <->  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 ) )
7 idd 21 . . . . . . . . 9  |-  ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  -> 
( p  =  q  ->  p  =  q ) )
87a1ii 24 . . . . . . . 8  |-  ( ph  ->  ( ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  ->  (
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  -> 
( p  =  q  ->  p  =  q ) ) ) )
98com14 82 . . . . . . 7  |-  ( p  =  q  ->  (
( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  -> 
( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  p  =  q ) ) ) )
10 isconcl5a.2 . . . . . . . . . . . . 13  |-  P  =  (PPoints `  F )
11 isconcl5a.1 . . . . . . . . . . . . 13  |-  L  =  (PLines `  F )
12 eqid 2283 . . . . . . . . . . . . 13  |-  ( line `  F )  =  (
line `  F )
13 isconcl5a.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e. Ig )
1413adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  F  e. Ig )
15 simpl31 1036 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  p  e.  P )
16 simpl21 1033 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  -> 
q  e.  P )
17 simpl1 958 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  p  =/=  q )
18 simpl32 1037 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  p  e.  L1 )
19 simpl22 1034 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  -> 
q  e.  L1 )
20 isconcl5a.4 . . . . . . . . . . . . . 14  |-  ( ph  -> 
L1  e.  L )
2120adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L1  e.  L )
2210, 11, 12, 14, 15, 16, 17, 18, 19, 21lineval42 26080 . . . . . . . . . . . 12  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L1  =  ( p (
line `  F )
q ) )
23 simpl33 1038 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  p  e.  L 2
)
24 simpl23 1035 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  -> 
q  e.  L 2
)
25 isconcl5a.5 . . . . . . . . . . . . . 14  |-  ( ph  ->  L 2  e.  L
)
2625adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L 2  e.  L
)
2710, 11, 12, 14, 15, 16, 17, 23, 24, 26lineval42 26080 . . . . . . . . . . . 12  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L 2  =  (
p ( line `  F
) q ) )
2822, 27eqtr4d 2318 . . . . . . . . . . 11  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L1  =  L 2 )
29 isconcl5a.6 . . . . . . . . . . . . 13  |-  ( ph  -> 
L1  =/=  L 2
)
3029neneqd 2462 . . . . . . . . . . . 12  |-  ( ph  ->  -.  L1  =  L 2 )
3130adantl 452 . . . . . . . . . . 11  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  -.  L1  =  L 2
)
3228, 31jca 518 . . . . . . . . . 10  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  -> 
( L1  =  L 2  /\  -.  L1  =  L 2 ) )
3332ex 423 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  ( ph  ->  (
L1  =  L 2  /\  -.  L1  =  L 2 ) ) )
34 pm2.24 101 . . . . . . . . . 10  |-  ( L1  =  L 2  ->  ( -.  L1  =  L 2  ->  p  =  q ) )
3534imp 418 . . . . . . . . 9  |-  ( (
L1  =  L 2  /\  -.  L1  =  L 2 )  ->  p  =  q )
3633, 35syl6 29 . . . . . . . 8  |-  ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  ( ph  ->  p  =  q ) )
37363exp 1150 . . . . . . 7  |-  ( p  =/=  q  ->  (
( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  -> 
( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  p  =  q ) ) ) )
389, 37pm2.61ine 2522 . . . . . 6  |-  ( ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  -> 
( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  p  =  q ) ) )
396, 38sylbi 187 . . . . 5  |-  ( [. q  /  p ]. (
p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  -> 
( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  p  =  q ) ) )
4039impcom 419 . . . 4  |-  ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  ( ph  ->  p  =  q ) )
4140com12 27 . . 3  |-  ( ph  ->  ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
4241alrimivv 1618 . 2  |-  ( ph  ->  A. p A. q
( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
43 nfv 1605 . . . 4  |-  F/ q ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )
4443mo3 2174 . . 3  |-  ( E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  <->  A. p A. q ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [ q  /  p ]
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
45 sbsbc 2995 . . . . . 6  |-  ( [ q  /  p ]
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  <->  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
4645anbi2i 675 . . . . 5  |-  ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [ q  /  p ]
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  <-> 
( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) ) )
4746imbi1i 315 . . . 4  |-  ( ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [
q  /  p ]
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q )  <->  ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
48472albii 1554 . . 3  |-  ( A. p A. q ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [ q  /  p ]
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q )  <->  A. p A. q
( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
4944, 48bitri 240 . 2  |-  ( E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  <->  A. p A. q ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
5042, 49sylibr 203 1  |-  ( ph  ->  E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623   [wsb 1629    e. wcel 1684   E*wmo 2144    =/= wne 2446   [.wsbc 2991   ` cfv 5255  (class class class)co 5858  PPointscpoints 26056  PLinescplines 26058  Igcig 26060   linecline 26076
This theorem is referenced by:  isconcl6a  26103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-ig2 26061  df-li 26077
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