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Theorem isconcl6ab 26104
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 26102 . 2  |-  ( ph  ->  E* p ( p  e.  L1  /\  p  e.  L 2 ) )
8 df-mo 2148 . . 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 3464 . . . . . 6  |-  ( (
L1  i^i  L 2
)  =/=  (/)  <->  E. p  p  e.  ( L1  i^i  L 2 ) )
11 elin 3358 . . . . . . . . 9  |-  ( p  e.  ( L1  i^i  L 2 )  <->  ( p  e.  L1  /\  p  e.  L 2 ) )
1211biimpi 186 . . . . . . . 8  |-  ( p  e.  ( L1  i^i  L 2 )  ->  (
p  e.  L1  /\  p  e.  L 2 ) )
1312eximi 1563 . . . . . . 7  |-  ( E. p  p  e.  (
L1  i^i  L 2
)  ->  E. p
( p  e.  L1  /\  p  e.  L 2
) )
1413a1d 22 . . . . . 6  |-  ( E. p  p  e.  (
L1  i^i  L 2
)  ->  ( ph  ->  E. p ( p  e.  L1  /\  p  e.  L 2 ) ) )
1510, 14sylbi 187 . . . . 5  |-  ( (
L1  i^i  L 2
)  =/=  (/)  ->  ( ph  ->  E. p ( p  e.  L1  /\  p  e.  L 2 ) ) )
169, 15mpcom 32 . . . 4  |-  ( ph  ->  E. p ( p  e.  L1  /\  p  e.  L 2 ) )
1716imim1i 54 . . 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 187 . 2  |-  ( E* p ( p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) ) )
197, 18mpcom 32 1  |-  ( ph  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   E*wmo 2144    =/= wne 2446    i^i cin 3151   (/)c0 3455   ` cfv 5255  PPointscpoints 26056  PLinescplines 26058  Igcig 26060
This theorem is referenced by:  isconcl7a  26105
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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