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Theorem mnfltxr 13166
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 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)

Proof of Theorem mnfltxr
StepHypRef Expression
1 mnflt 13162 . 2 (𝐴 ∈ ℝ → -∞ < 𝐴)
2 mnfltpnf 13165 . . 3 -∞ < +∞
3 breq2 5151 . . 3 (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞))
42, 3mpbiri 258 . 2 (𝐴 = +∞ → -∞ < 𝐴)
51, 4jaoi 857 1 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 847   = wceq 1536  wcel 2105   class class class wbr 5147  cr 11151  +∞cpnf 11289  -∞cmnf 11290   < clt 11292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297
This theorem is referenced by:  supxrgtmnf  13367  nmogtmnf  30798  nmopgtmnf  31896
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