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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 PLines
isconcl5a.2 PPoints
isconcl5a.3 Ig
isconcl5a.4
isconcl5a.5
isconcl5a.6
isconcl6ab.1
Assertion
Ref Expression
isconcl6ab
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem isconcl6ab
StepHypRef Expression
1 isconcl5a.1 . . 3 PLines
2 isconcl5a.2 . . 3 PPoints
3 isconcl5a.3 . . 3 Ig
4 isconcl5a.4 . . 3
5 isconcl5a.5 . . 3
6 isconcl5a.6 . . 3
71, 2, 3, 4, 5, 6isconcl5ab 25268 . 2
8 df-mo 2119 . . 3
9 isconcl6ab.1 . . . . 5
10 n0 3371 . . . . . 6
11 elin 3266 . . . . . . . . 9
1211biimpi 188 . . . . . . . 8
1312eximi 1574 . . . . . . 7
1413a1d 24 . . . . . 6
1510, 14sylbi 189 . . . . 5
169, 15mpcom 34 . . . 4
1716imim1i 56 . . 3
188, 17sylbi 189 . 2
197, 18mpcom 34 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360  wex 1537   wceq 1619   wcel 1621  weu 2114  wmo 2115   wne 2412   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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