Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isibg1a5a Unicode version

Theorem isibg1a5a 26227
 Description: If is between and , then and , are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
isibg1a3a.1 PPoints
isibg1a3a.2 btw
isibg1a3a.3 Ibg
isibg1a3a.4
isibg1a3a.5
isibg1a3a.6
isibg1a3a.7
Assertion
Ref Expression
isibg1a5a

Proof of Theorem isibg1a5a
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isibg1a3a.3 . 2 Ibg
2 isibg1a3a.1 . . . 4 PPoints
3 eqid 2296 . . . 4 PLines PLines
4 isibg1a3a.2 . . . 4 btw
5 eqid 2296 . . . 4 coln coln
62, 3, 4, 5isibg2 26213 . . 3 Ibg Ig coln coln PLines
7 isibg1a3a.4 . . . . . . 7
8 isibg1a3a.5 . . . . . . 7
92, 3isibg2aalem1 26216 . . . . . . . . 9 coln coln PLines coln coln PLines
10 isibg1a3a.6 . . . . . . . . . . . 12
11 isibg2aalem2 26217 . . . . . . . . . . . . . 14 coln coln PLines coln coln PLines
12 isibg1a3a.7 . . . . . . . . . . . . . . . 16
13 simp33 993 . . . . . . . . . . . . . . . 16 coln
1412, 13imim12i 53 . . . . . . . . . . . . . . 15 coln
1514ad2antrl 708 . . . . . . . . . . . . . 14 coln coln PLines
1611, 15syl6 29 . . . . . . . . . . . . 13 coln coln PLines
1716com23 72 . . . . . . . . . . . 12 coln coln PLines
1810, 17mpcom 32 . . . . . . . . . . 11 coln coln PLines
1918com12 27 . . . . . . . . . 10 coln coln PLines
2019adantl 452 . . . . . . . . 9 coln coln PLines
219, 20syl6 29 . . . . . . . 8 coln coln PLines
2221com23 72 . . . . . . 7 coln coln PLines
237, 8, 22syl2anc 642 . . . . . 6 coln coln PLines
2423pm2.43i 43 . . . . 5 coln coln PLines
2524com12 27 . . . 4 coln coln PLines
2625adantl 452 . . 3 Ig coln coln PLines
276, 26sylbi 187 . 2 Ibg
281, 27mpcom 32 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3o 933   w3a 934   wceq 1632   wcel 1696   wne 2459   wnel 2460  wral 2556  wrex 2557   cin 3164  c0 3468  ctp 3655  cfv 5271  (class class class)co 5874  PPointscpoints 26159  PLinescplines 26161  Igcig 26163  colnccol 26193  btwcbtw 26209  Ibgcibg 26210 This theorem is referenced by:  isibg1a7  26229  lppotos  26247 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-ibg2 26212
 Copyright terms: Public domain W3C validator