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Theorem mnfltxr 13047
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 13043 . 2 (𝐴 ∈ ℝ → -∞ < 𝐴)
2 mnfltpnf 13046 . . 3 -∞ < +∞
3 breq2 5109 . . 3 (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞))
42, 3mpbiri 257 . 2 (𝐴 = +∞ → -∞ < 𝐴)
51, 4jaoi 855 1 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 845   = wceq 1541  wcel 2106   class class class wbr 5105  cr 11049  +∞cpnf 11185  -∞cmnf 11186   < clt 11188
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671  ax-cnex 11106
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-xp 5639  df-pnf 11190  df-mnf 11191  df-xr 11192  df-ltxr 11193
This theorem is referenced by:  supxrgtmnf  13247  nmogtmnf  29659  nmopgtmnf  30757
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