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Theorem ltlecasei 8944
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 8943 . . 3  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <_  A  \/  A  <  B ) )
63, 4, 5syl2anc 642 . 2  |-  ( ph  ->  ( B  <_  A  \/  A  <  B ) )
71, 2, 6mpjaodan 761 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    e. wcel 1696   class class class wbr 4039   RRcr 8752    < clt 8883    <_ cle 8884
This theorem is referenced by:  iccsplit  10784  expnbnd  11246  hashf1  11411  absmax  11829  sinltx  12485  iccntr  18342  pmltpclem2  18825  cniccbdd  18837  iccvolcl  18940  dyaddisjlem  18966  mbfposr  19023  itg1ge0a  19082  itg2monolem1  19121  itgioo  19186  c1lip1  19360  plyeq0lem  19608  aalioulem5  19732  pserulm  19814  tanord  19916  birthdaylem3  20264  fsumharmonic  20321  chpo1ubb  20646  ioovolcl  27845
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  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-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-xr 8887  df-le 8889
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