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Theorem isconcl6ab 25457
Description: Two distinct non-parallel lines intersect in one and only 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
)
isconcl6ab.1  |-  ( ph  ->  ( L1  i^i  L 2 )  =/=  (/) )
Assertion
Ref Expression
isconcl6ab  |-  ( 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 isconcl6ab
StepHypRef Expression
1 isconcl5a.1 . . 3  |-  L  =  (PLines `  F )
2 isconcl5a.2 . . 3  |-  P  =  (PPoints `  F )
3 isconcl5a.3 . . 3  |-  ( ph  ->  F  e. Ig )
4 isconcl5a.4 . . 3  |-  ( ph  -> 
L1  e.  L )
5 isconcl5a.5 . . 3  |-  ( ph  ->  L 2  e.  L
)
6 isconcl5a.6 . . 3  |-  ( ph  -> 
L1  =/=  L 2
)
71, 2, 3, 4, 5, 6isconcl5ab 25455 . 2  |-  ( ph  ->  E* p ( p  e.  L1  /\  p  e.  L 2 ) )
8 df-mo 2122 . . 3  |-  ( E* p ( p  e.  L1  /\  p  e.  L 2 )  <->  ( E. p ( p  e.  L1  /\  p  e.  L 2 )  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) ) )
9 isconcl6ab.1 . . . . 5  |-  ( ph  ->  ( L1  i^i  L 2 )  =/=  (/) )
10 n0 3425 . . . . . 6  |-  ( (
L1  i^i  L 2
)  =/=  (/)  <->  E. p  p  e.  ( L1  i^i  L 2 ) )
11 elin 3319 . . . . . . . . 9  |-  ( p  e.  ( L1  i^i  L 2 )  <->  ( p  e.  L1  /\  p  e.  L 2 ) )
1211biimpi 188 . . . . . . . 8  |-  ( p  e.  ( L1  i^i  L 2 )  ->  (
p  e.  L1  /\  p  e.  L 2 ) )
1312eximi 1574 . . . . . . 7  |-  ( E. p  p  e.  (
L1  i^i  L 2
)  ->  E. p
( p  e.  L1  /\  p  e.  L 2
) )
1413a1d 24 . . . . . 6  |-  ( E. p  p  e.  (
L1  i^i  L 2
)  ->  ( ph  ->  E. p ( p  e.  L1  /\  p  e.  L 2 ) ) )
1510, 14sylbi 189 . . . . 5  |-  ( (
L1  i^i  L 2
)  =/=  (/)  ->  ( ph  ->  E. p ( p  e.  L1  /\  p  e.  L 2 ) ) )
169, 15mpcom 34 . . . 4  |-  ( ph  ->  E. p ( p  e.  L1  /\  p  e.  L 2 ) )
1716imim1i 56 . . 3  |-  ( ( E. p ( p  e.  L1  /\  p  e.  L 2 )  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) )  -> 
( ph  ->  E! p
( p  e.  L1  /\  p  e.  L 2
) ) )
188, 17sylbi 189 . 2  |-  ( E* p ( p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) ) )
197, 18mpcom 34 1  |-  ( ph  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   E!weu 2117   E*wmo 2118    =/= wne 2419    i^i cin 3112   (/)c0 3416   ` cfv 4659  PPointscpoints 25409  PLinescplines 25411  Igcig 25413
This theorem is referenced by:  isconcl7a  25458
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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