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Theorem mnfltxr 10617
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 10615 . 2  |-  ( A  e.  RR  ->  -oo  <  A )
2 mnfltpnf 10616 . . 3  |-  -oo  <  +oo
3 breq2 4129 . . 3  |-  ( A  =  +oo  ->  (  -oo  <  A  <->  -oo  <  +oo ) )
42, 3mpbiri 224 . 2  |-  ( A  =  +oo  ->  -oo  <  A )
51, 4jaoi 368 1  |-  ( ( A  e.  RR  \/  A  =  +oo )  ->  -oo  <  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    = wceq 1647    e. wcel 1715   class class class wbr 4125   RRcr 8883    +oocpnf 9011    -oocmnf 9012    < clt 9014
This theorem is referenced by:  supxrgtmnf  10801  nmogtmnf  21782  nmopgtmnf  22882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-xp 4798  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019
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