Theorem mnfltxr 13076

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 13072	. 2 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
2		mnfltpnf 13075	. . 3 ⊢ -∞ < +∞
3		breq2 5083	. . 3 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞))
4	2, 3	mpbiri 259	. 2 ⊢ (𝐴 = +∞ → -∞ < 𝐴)
5	1, 4	jaoi 863	1 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)

Syntax hints:   → wi 4   ∨ wo 853   = wceq 1547   ∈ wcel 2119   class class class wbr 5079  ℝcr 11035  +∞cpnf 11174  -∞cmnf 11175   < clt 11177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5631  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182

This theorem is referenced by:  supxrgtmnf  13279  nmogtmnf  30866  nmopgtmnf  31964