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Theorem mnfltxr 10755
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 10753 . 2  |-  ( A  e.  RR  ->  -oo  <  A )
2 mnfltpnf 10754 . . 3  |-  -oo  <  +oo
3 breq2 4241 . . 3  |-  ( A  =  +oo  ->  (  -oo  <  A  <->  -oo  <  +oo ) )
42, 3mpbiri 226 . 2  |-  ( A  =  +oo  ->  -oo  <  A )
51, 4jaoi 370 1  |-  ( ( A  e.  RR  \/  A  =  +oo )  ->  -oo  <  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    = wceq 1653    e. wcel 1727   class class class wbr 4237   RRcr 9020    +oocpnf 9148    -oocmnf 9149    < clt 9151
This theorem is referenced by:  supxrgtmnf  10939  nmogtmnf  22302  nmopgtmnf  23402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-xp 4913  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156
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