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Theorem isconcl6ab 25270
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 25268 . 2  |-  ( ph  ->  E* p ( p  e.  L1  /\  p  e.  L 2 ) )
8 df-mo 2119 . . 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 3371 . . . . . 6  |-  ( (
L1  i^i  L 2
)  =/=  (/)  <->  E. p  p  e.  ( L1  i^i  L 2 ) )
11 elin 3266 . . . . . . . . 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 2114   E*wmo 2115    =/= wne 2412    i^i cin 3077   (/)c0 3362   ` cfv 4592  PPointscpoints 25222  PLinescplines 25224  Igcig 25226
This theorem is referenced by:  isconcl7a  25271
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-ig2 25227  df-li 25243
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