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Theorem ltlecasei 9173
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 9172 . . 3  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <_  A  \/  A  <  B ) )
63, 4, 5syl2anc 643 . 2  |-  ( ph  ->  ( B  <_  A  \/  A  <  B ) )
71, 2, 6mpjaodan 762 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    e. wcel 1725   class class class wbr 4204   RRcr 8981    < clt 9112    <_ cle 9113
This theorem is referenced by:  iccsplit  11021  expnbnd  11500  hashf1  11698  absmax  12125  sinltx  12782  iccntr  18844  pmltpclem2  19338  cniccbdd  19350  iccvolcl  19453  dyaddisjlem  19479  mbfposr  19536  itg1ge0a  19595  itg2monolem1  19634  itgioo  19699  c1lip1  19873  plyeq0lem  20121  aalioulem5  20245  pserulm  20330  tanord  20432  birthdaylem3  20784  fsumharmonic  20842  chpo1ubb  21167  mblfinlem  26234  ioovolcl  27709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-xr 9116  df-le 9118
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