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Theorem mnfltpnf 11904
Description: Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
mnfltpnf -∞ < +∞

Proof of Theorem mnfltpnf
StepHypRef Expression
1 eqid 2621 . . . 4 -∞ = -∞
2 eqid 2621 . . . 4 +∞ = +∞
3 olc 399 . . . 4 ((-∞ = -∞ ∧ +∞ = +∞) → (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ < +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)))
41, 2, 3mp2an 707 . . 3 (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ < +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞))
54orci 405 . 2 ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ < +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))
6 mnfxr 10040 . . 3 -∞ ∈ ℝ*
7 pnfxr 10036 . . 3 +∞ ∈ ℝ*
8 ltxr 11893 . . 3 ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ < +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))))
96, 7, 8mp2an 707 . 2 (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ < +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ))))
105, 9mpbir 221 1 -∞ < +∞
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987   class class class wbr 4613  cr 9879   < cltrr 9884  +∞cpnf 10015  -∞cmnf 10016  *cxr 10017   < clt 10018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023
This theorem is referenced by:  mnfltxr  11905  xrlttri  11916  xrlttr  11917  xltnegi  11990
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