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Theorem ngtmnft 8481
 Description: An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)
Assertion
Ref Expression
ngtmnft (A * → (A = -∞ ↔ ¬ -∞ < A))

Proof of Theorem ngtmnft
StepHypRef Expression
1 elxr 8446 . 2 (A * ↔ (A A = +∞ A = -∞))
2 renemnf 6851 . . . . 5 (A ℝ → A ≠ -∞)
32neneqd 2221 . . . 4 (A ℝ → ¬ A = -∞)
4 mnflt 8454 . . . . 5 (A ℝ → -∞ < A)
5 notnot1 559 . . . . 5 (-∞ < A → ¬ ¬ -∞ < A)
64, 5syl 14 . . . 4 (A ℝ → ¬ ¬ -∞ < A)
73, 62falsed 617 . . 3 (A ℝ → (A = -∞ ↔ ¬ -∞ < A))
8 pnfnemnf 8447 . . . . . 6 +∞ ≠ -∞
9 neeq1 2213 . . . . . 6 (A = +∞ → (A ≠ -∞ ↔ +∞ ≠ -∞))
108, 9mpbiri 157 . . . . 5 (A = +∞ → A ≠ -∞)
1110neneqd 2221 . . . 4 (A = +∞ → ¬ A = -∞)
12 mnfltpnf 8456 . . . . . . 7 -∞ < +∞
13 breq2 3759 . . . . . . 7 (A = +∞ → (-∞ < A ↔ -∞ < +∞))
1412, 13mpbiri 157 . . . . . 6 (A = +∞ → -∞ < A)
1514necon3bi 2249 . . . . 5 (¬ -∞ < AA ≠ +∞)
1615necon2bi 2254 . . . 4 (A = +∞ → ¬ ¬ -∞ < A)
1711, 162falsed 617 . . 3 (A = +∞ → (A = -∞ ↔ ¬ -∞ < A))
18 id 19 . . . 4 (A = -∞ → A = -∞)
19 mnfxr 8444 . . . . . 6 -∞ *
20 xrltnr 8451 . . . . . 6 (-∞ * → ¬ -∞ < -∞)
2119, 20ax-mp 7 . . . . 5 ¬ -∞ < -∞
22 breq2 3759 . . . . 5 (A = -∞ → (-∞ < A ↔ -∞ < -∞))
2321, 22mtbiri 599 . . . 4 (A = -∞ → ¬ -∞ < A)
2418, 232thd 164 . . 3 (A = -∞ → (A = -∞ ↔ ¬ -∞ < A))
257, 17, 243jaoi 1197 . 2 ((A A = +∞ A = -∞) → (A = -∞ ↔ ¬ -∞ < A))
261, 25sylbi 114 1 (A * → (A = -∞ ↔ ¬ -∞ < A))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ w3o 883   = wceq 1242   ∈ wcel 1390   ≠ wne 2201   class class class wbr 3755  ℝcr 6690  +∞cpnf 6834  -∞cmnf 6835  ℝ*cxr 6836   < clt 6837 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6754  ax-resscn 6755  ax-pre-ltirr 6775 This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6839  df-mnf 6840  df-xr 6841  df-ltxr 6842 This theorem is referenced by:  ge0nemnf  8487
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