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Theorem mnfltxr 13107
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 13103 . 2 (𝐴 ∈ ℝ → -∞ < 𝐴)
2 mnfltpnf 13106 . . 3 -∞ < +∞
3 breq2 5153 . . 3 (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞))
42, 3mpbiri 258 . 2 (𝐴 = +∞ → -∞ < 𝐴)
51, 4jaoi 856 1 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846   = wceq 1542  wcel 2107   class class class wbr 5149  cr 11109  +∞cpnf 11245  -∞cmnf 11246   < clt 11248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253
This theorem is referenced by:  supxrgtmnf  13308  nmogtmnf  30023  nmopgtmnf  31121
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