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Theorem mnfltxr 13111
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 13107 . 2 (𝐴 ∈ ℝ → -∞ < 𝐴)
2 mnfltpnf 13110 . . 3 -∞ < +∞
3 breq2 5151 . . 3 (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞))
42, 3mpbiri 257 . 2 (𝐴 = +∞ → -∞ < 𝐴)
51, 4jaoi 853 1 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 843   = wceq 1539  wcel 2104   class class class wbr 5147  cr 11111  +∞cpnf 11249  -∞cmnf 11250   < clt 11252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257
This theorem is referenced by:  supxrgtmnf  13312  nmogtmnf  30290  nmopgtmnf  31388
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