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Theorem mnfltxr 12174
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 12170 . 2 (𝐴 ∈ ℝ → -∞ < 𝐴)
2 mnfltpnf 12173 . . 3 -∞ < +∞
3 breq2 4808 . . 3 (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞))
42, 3mpbiri 248 . 2 (𝐴 = +∞ → -∞ < 𝐴)
51, 4jaoi 393 1 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382   = wceq 1632  wcel 2139   class class class wbr 4804  cr 10147  +∞cpnf 10283  -∞cmnf 10284   < clt 10286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291
This theorem is referenced by:  supxrgtmnf  12372  nmogtmnf  27955  nmopgtmnf  29057
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