Theorem mnfltxr 13069

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 13065	. 2 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
2		mnfltpnf 13068	. . 3 ⊢ -∞ < +∞
3		breq2 5090	. . 3 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞))
4	2, 3	mpbiri 258	. 2 ⊢ (𝐴 = +∞ → -∞ < 𝐴)
5	1, 4	jaoi 858	1 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)

Syntax hints:   → wi 4   ∨ wo 848   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  ℝcr 11028  +∞cpnf 11167  -∞cmnf 11168   < clt 11170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175

This theorem is referenced by:  supxrgtmnf  13272  nmogtmnf  30856  nmopgtmnf  31954