Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iscola2 Unicode version

Theorem iscola2 26092
 Description: The predicate "being collinear points". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypothesis
Ref Expression
iscola2.1 Ig
Assertion
Ref Expression
iscola2 coln PLines
Distinct variable group:   ,,
Allowed substitution hints:   (,)

Proof of Theorem iscola2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 iscola2.1 . 2 Ig
2 iunab 3948 . . 3 PLines PLines
3 fvex 5539 . . . . 5 PLines
43a1i 10 . . . 4 PLines
5 pm4.24 624 . . . . . . 7
65abbii 2395 . . . . . 6
7 abssexg 4195 . . . . . 6 PLines
86, 7syl5eqel 2367 . . . . 5 PLines
98rgen 2608 . . . 4 PLines
10 iunexg 5767 . . . 4 PLines PLines PLines
114, 9, 10sylancl 643 . . 3 PLines
122, 11syl5eqelr 2368 . 2 PLines
13 fveq2 5525 . . . . 5 PLines PLines
1413rexeqdv 2743 . . . 4 PLines PLines
1514abbidv 2397 . . 3 PLines PLines
16 df-col 26091 . . 3 coln Ig PLines
1715, 16fvmptg 5600 . 2 Ig PLines coln PLines
181, 12, 17syl2anc 642 1 coln PLines
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684  cab 2269  wral 2543  wrex 2544  cvv 2788   wss 3152  ciun 3905  cfv 5255  PLinescplines 26058  Igcig 26060  colnccol 26090 This theorem is referenced by:  iscol2  26093 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-col 26091
 Copyright terms: Public domain W3C validator