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Theorem ltlecasei 9741
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 9740 . . 3  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <_  A  \/  A  <  B ) )
63, 4, 5syl2anc 665 . 2  |-  ( ph  ->  ( B  <_  A  \/  A  <  B ) )
71, 2, 6mpjaodan 793 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    e. wcel 1870   class class class wbr 4426   RRcr 9537    < clt 9674    <_ cle 9675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-cnv 4862  df-xr 9678  df-le 9680
This theorem is referenced by:  iccsplit  11763  expnbnd  12398  hashf1  12615  absmax  13371  sinltx  14221  iccntr  21750  pmltpclem2  22281  cniccbdd  22293  iccvolcl  22397  ioovolcl  22399  dyaddisjlem  22430  mbfposr  22485  itg1ge0a  22546  itg2monolem1  22585  itgioo  22650  c1lip1  22826  plyeq0lem  23032  aalioulem5  23157  pserulm  23242  tanord  23352  birthdaylem3  23744  fsumharmonic  23802  chpo1ubb  24182  mblfinlem2  31682  ioodvbdlimc1  37377  ioodvbdlimc2  37379  ibliooicc  37417  fourierdlem107  37645
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