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Theorem mnfltxr 10688
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 10686 . 2  |-  ( A  e.  RR  ->  -oo  <  A )
2 mnfltpnf 10687 . . 3  |-  -oo  <  +oo
3 breq2 4184 . . 3  |-  ( A  =  +oo  ->  (  -oo  <  A  <->  -oo  <  +oo ) )
42, 3mpbiri 225 . 2  |-  ( A  =  +oo  ->  -oo  <  A )
51, 4jaoi 369 1  |-  ( ( A  e.  RR  \/  A  =  +oo )  ->  -oo  <  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    = wceq 1649    e. wcel 1721   class class class wbr 4180   RRcr 8953    +oocpnf 9081    -oocmnf 9082    < clt 9084
This theorem is referenced by:  supxrgtmnf  10872  nmogtmnf  22232  nmopgtmnf  23332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-xp 4851  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089
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