ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xrltne Unicode version

Theorem xrltne 9276
Description: 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
Assertion
Ref Expression
xrltne  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  =/=  A )

Proof of Theorem xrltne
StepHypRef Expression
1 xrltnr 9248 . . . . 5  |-  ( A  e.  RR*  ->  -.  A  <  A )
2 breq2 3849 . . . . . 6  |-  ( B  =  A  ->  ( A  <  B  <->  A  <  A ) )
32notbid 627 . . . . 5  |-  ( B  =  A  ->  ( -.  A  <  B  <->  -.  A  <  A ) )
41, 3syl5ibrcom 155 . . . 4  |-  ( A  e.  RR*  ->  ( B  =  A  ->  -.  A  <  B ) )
54necon2ad 2312 . . 3  |-  ( A  e.  RR*  ->  ( A  <  B  ->  B  =/=  A ) )
65imp 122 . 2  |-  ( ( A  e.  RR*  /\  A  <  B )  ->  B  =/=  A )
763adant2 962 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  =/=  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 924    = wceq 1289    e. wcel 1438    =/= wne 2255   class class class wbr 3845   RR*cxr 7519    < clt 7520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-pre-ltirr 7455
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator