Theorem mnfltxr 13148

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

Step	Hyp	Ref	Expression
1		mnflt 13144	. 2 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
2		mnfltpnf 13147	. . 3 ⊢ -∞ < +∞
3		breq2 5114	. . 3 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞))
4	2, 3	mpbiri 261	. 2 ⊢ (𝐴 = +∞ → -∞ < 𝐴)
5	1, 4	jaoi 870	1 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1567   ∈ wcel 2149   class class class wbr 5110  ℝcr 11095  +∞cpnf 11236  -∞cmnf 11237   < clt 11239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5665  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244

This theorem is referenced by:  supxrgtmnf  13351  nmogtmnf  31059  nmopgtmnf  32157