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Theorem isconcl5a 25500
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
Dummy variable  q is distinct from all other variables.
Allowed substitution hints:    F( p)    L( p)

Proof of Theorem isconcl5a
StepHypRef Expression
1 vex 2792 . . . . . . 7  |-  q  e. 
_V
2 eleq1 2344 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  P  <->  q  e.  P ) )
3 eleq1 2344 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  L1  <->  q  e.  L1 ) )
4 eleq1 2344 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  L 2  <->  q  e.  L 2 )
)
52, 3, 43anbi123d 1254 . . . . . . 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 3026 . . . . . 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 23 . . . . . . . . 9  |-  ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  -> 
( p  =  q  ->  p  =  q ) )
87a1ii 26 . . . . . . . 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 84 . . . . . . 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 2284 . . . . . . . . . . . . 13  |-  ( line `  F )  =  (
line `  F )
13 isconcl5a.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e. Ig )
1413adantl 454 . . . . . . . . . . . . 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 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.  P )
16 simpl21 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.  P )
17 simpl1 960 . . . . . . . . . . . . 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 1039 . . . . . . . . . . . . 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 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 )  -> 
q  e.  L1 )
20 isconcl5a.4 . . . . . . . . . . . . . 14  |-  ( ph  -> 
L1  e.  L )
2120adantl 454 . . . . . . . . . . . . 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 25479 . . . . . . . . . . . 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 1040 . . . . . . . . . . . . 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 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 )  -> 
q  e.  L 2
)
25 isconcl5a.5 . . . . . . . . . . . . . 14  |-  ( ph  ->  L 2  e.  L
)
2625adantl 454 . . . . . . . . . . . . 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 25479 . . . . . . . . . . . 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 2319 . . . . . . . . . . 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 2463 . . . . . . . . . . . 12  |-  ( ph  ->  -.  L1  =  L 2 )
3130adantl 454 . . . . . . . . . . 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 520 . . . . . . . . . 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 425 . . . . . . . . 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 pm2.24 103 . . . . . . . . . 10  |-  ( L1  =  L 2  ->  ( -.  L1  =  L 2  ->  p  =  q ) )
3534imp 420 . . . . . . . . 9  |-  ( (
L1  =  L 2  /\  -.  L1  =  L 2 )  ->  p  =  q )
3633, 35syl6 31 . . . . . . . 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 1152 . . . . . . 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 2523 . . . . . 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 189 . . . . 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 421 . . . 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 29 . . 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 1619 . 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 1606 . . . 4  |-  F/ q ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )
4443mo3 2175 . . 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 2996 . . . . . 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 677 . . . . 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 317 . . . 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 1555 . . 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 242 . 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 205 1  |-  ( ph  ->  E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 936   A.wal 1528    = wceq 1624   [wsb 1631    e. wcel 1685   E*wmo 2145    =/= wne 2447   [.wsbc 2992   ` cfv 5221  (class class class)co 5819  PPointscpoints 25455  PLinescplines 25457  Igcig 25459   linecline 25475
This theorem is referenced by:  isconcl6a  25502
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-ig2 25460  df-li 25476
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