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Theorem isconcl5a 25454
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
StepHypRef Expression
1 vex 2760 . . . . . . 7  |-  q  e. 
_V
2 eleq1 2316 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  P  <->  q  e.  P ) )
3 eleq1 2316 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  L1  <->  q  e.  L1 ) )
4 eleq1 2316 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  L 2  <->  q  e.  L 2 )
)
52, 3, 43anbi123d 1257 . . . . . . 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 2986 . . . . . 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 2256 . . . . . . . . . . . . 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 1041 . . . . . . . . . . . . 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 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 )  -> 
q  e.  P )
17 simpl1 963 . . . . . . . . . . . . 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 1042 . . . . . . . . . . . . 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 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 )  -> 
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 25433 . . . . . . . . . . . 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 1043 . . . . . . . . . . . . 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 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 )  -> 
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 25433 . . . . . . . . . . . 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 2291 . . . . . . . . . . 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 2435 . . . . . . . . . . . 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 1155 . . . . . . 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 2495 . . . . . 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 2014 . 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 1629 . . . 4  |-  F/ q ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )
4443mo3 2147 . . 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 2956 . . . . . 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 678 . . . . 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 939   A.wal 1532    = wceq 1619    e. wcel 1621   [wsb 1883   E*wmo 2118    =/= wne 2419   [.wsbc 2952   ` cfv 4659  (class class class)co 5778  PPointscpoints 25409  PLinescplines 25411  Igcig 25413   linecline 25429
This theorem is referenced by:  isconcl6a  25456
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 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-ig2 25414  df-li 25430
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