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Theorem mnfltxr 8989
Description: Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)
Assertion
Ref Expression
mnfltxr  |-  ( ( A  e.  RR  \/  A  = +oo )  -> -oo  <  A )

Proof of Theorem mnfltxr
StepHypRef Expression
1 mnflt 8986 . 2  |-  ( A  e.  RR  -> -oo  <  A )
2 mnfltpnf 8988 . . 3  |- -oo  < +oo
3 breq2 3809 . . 3  |-  ( A  = +oo  ->  ( -oo  <  A  <-> -oo  < +oo ) )
42, 3mpbiri 166 . 2  |-  ( A  = +oo  -> -oo  <  A )
51, 4jaoi 669 1  |-  ( ( A  e.  RR  \/  A  = +oo )  -> -oo  <  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662    = wceq 1285    e. wcel 1434   class class class wbr 3805   RRcr 7094   +oocpnf 7264   -oocmnf 7265    < clt 7267
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-cnex 7181
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272
This theorem is referenced by:  xrltso  8999
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