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Theorem ltlecasei 9692
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 9691 . . 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 1767   class class class wbr 4447   RRcr 9491    < clt 9628    <_ cle 9629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-xr 9632  df-le 9634
This theorem is referenced by:  iccsplit  11653  expnbnd  12263  hashf1  12472  absmax  13125  sinltx  13785  iccntr  21089  pmltpclem2  21624  cniccbdd  21636  iccvolcl  21740  ioovolcl  21742  dyaddisjlem  21767  mbfposr  21822  itg1ge0a  21881  itg2monolem1  21920  itgioo  21985  c1lip1  22161  plyeq0lem  22370  aalioulem5  22494  pserulm  22579  tanord  22686  birthdaylem3  23039  fsumharmonic  23097  chpo1ubb  23422  mblfinlem2  29657  ibliooicc  31317
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