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Theorem ltlen 9675
Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.)
Assertion
Ref Expression
ltlen  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( A  <_  B  /\  B  =/=  A ) ) )

Proof of Theorem ltlen
StepHypRef Expression
1 ltle 9662 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <_  B )
)
2 ltneOLD 9671 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  =/=  A )
323expia 1196 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  B  =/=  A ) )
41, 3jcad 531 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  ( A  <_  B  /\  B  =/=  A
) ) )
5 leloe 9660 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  ( A  <  B  \/  A  =  B )
) )
6 ax-1 6 . . . . 5  |-  ( A  <  B  ->  ( B  =/=  A  ->  A  <  B ) )
7 df-ne 2651 . . . . . 6  |-  ( B  =/=  A  <->  -.  B  =  A )
8 pm2.24 109 . . . . . . 7  |-  ( B  =  A  ->  ( -.  B  =  A  ->  A  <  B ) )
98eqcoms 2466 . . . . . 6  |-  ( A  =  B  ->  ( -.  B  =  A  ->  A  <  B ) )
107, 9syl5bi 217 . . . . 5  |-  ( A  =  B  ->  ( B  =/=  A  ->  A  <  B ) )
116, 10jaoi 377 . . . 4  |-  ( ( A  <  B  \/  A  =  B )  ->  ( B  =/=  A  ->  A  <  B ) )
125, 11syl6bi 228 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( B  =/=  A  ->  A  <  B ) ) )
1312impd 429 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  <_  B  /\  B  =/=  A
)  ->  A  <  B ) )
144, 13impbid 191 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( A  <_  B  /\  B  =/=  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   RRcr 9480    < clt 9617    <_ cle 9618
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-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  df-le 9623
This theorem is referenced by:  ltleni  9691  ltlend  9719  nn0lt2  10923  rpneg  11251  fzofzim  11846  elfznelfzob  11897  hashsdom  12432  cnpart  13155  chfacfisf  19522  chfacfisfcpmat  19523  ang180lem2  23341  mumullem2  23652  lgsneg  23792  lgsdilem  23795  lgsdirprm  23802  axlowdimlem16  24462  unitdivcld  28118  itg2addnclem  30306  difmodm1lt  33389
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