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Theorem xrltne 11369
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 orc 383 . . . 4  |-  ( A  <  B  ->  ( A  <  B  \/  B  <  A ) )
2 xrltso 11350 . . . . . 6  |-  <  Or  RR*
3 sotrieq 4816 . . . . . 6  |-  ( (  <  Or  RR*  /\  ( A  e.  RR*  /\  B  e.  RR* ) )  -> 
( A  =  B  <->  -.  ( A  <  B  \/  B  <  A ) ) )
42, 3mpan 668 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  -.  ( A  <  B  \/  B  <  A ) ) )
54necon2abid 2708 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  B  \/  B  <  A )  <-> 
A  =/=  B ) )
61, 5syl5ib 219 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  A  =/=  B ) )
763impia 1191 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  =/=  B )
87necomd 2725 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  =/=  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439    Or wor 4788   RR*cxr 9616    < clt 9617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622
This theorem is referenced by:  xmulpnf1  11469  supxrbnd  11523  sgnp  13005  sgnn  13009  xrsdsreclblem  18659  supxrnemnf  27817  icoiccdif  31803  cncfiooicclem1  31935
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