MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltlecasei Structured version   Unicode version

Theorem ltlecasei 9487
Description: Ordering elimination by cases. (Contributed by NM, 1-Jul-2007.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
ltlecasei.1  |-  ( (
ph  /\  A  <  B )  ->  ps )
ltlecasei.2  |-  ( (
ph  /\  B  <_  A )  ->  ps )
ltlecasei.3  |-  ( ph  ->  A  e.  RR )
ltlecasei.4  |-  ( ph  ->  B  e.  RR )
Assertion
Ref Expression
ltlecasei  |-  ( ph  ->  ps )

Proof of Theorem ltlecasei
StepHypRef Expression
1 ltlecasei.2 . 2  |-  ( (
ph  /\  B  <_  A )  ->  ps )
2 ltlecasei.1 . 2  |-  ( (
ph  /\  A  <  B )  ->  ps )
3 ltlecasei.4 . . 3  |-  ( ph  ->  B  e.  RR )
4 ltlecasei.3 . . 3  |-  ( ph  ->  A  e.  RR )
5 lelttric 9486 . . 3  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <_  A  \/  A  <  B ) )
63, 4, 5syl2anc 661 . 2  |-  ( ph  ->  ( B  <_  A  \/  A  <  B ) )
71, 2, 6mpjaodan 784 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    e. wcel 1756   class class class wbr 4297   RRcr 9286    < clt 9423    <_ cle 9424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-cnv 4853  df-xr 9427  df-le 9429
This theorem is referenced by:  iccsplit  11423  expnbnd  11998  hashf1  12215  absmax  12822  sinltx  13478  iccntr  20403  pmltpclem2  20938  cniccbdd  20950  iccvolcl  21053  ioovolcl  21055  dyaddisjlem  21080  mbfposr  21135  itg1ge0a  21194  itg2monolem1  21233  itgioo  21298  c1lip1  21474  plyeq0lem  21683  aalioulem5  21807  pserulm  21892  tanord  21999  birthdaylem3  22352  fsumharmonic  22410  chpo1ubb  22735  mblfinlem2  28434
  Copyright terms: Public domain W3C validator