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Theorem gtnelicc 40521
Description: A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
gtnelicc.a (𝜑𝐴 ∈ ℝ*)
gtnelicc.b (𝜑𝐵 ∈ ℝ)
gtnelicc.c (𝜑𝐶 ∈ ℝ*)
gtnelicc.bltc (𝜑𝐵 < 𝐶)
Assertion
Ref Expression
gtnelicc (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Proof of Theorem gtnelicc
StepHypRef Expression
1 gtnelicc.bltc . . . 4 (𝜑𝐵 < 𝐶)
2 gtnelicc.b . . . . . 6 (𝜑𝐵 ∈ ℝ)
32rexrd 10406 . . . . 5 (𝜑𝐵 ∈ ℝ*)
4 gtnelicc.c . . . . 5 (𝜑𝐶 ∈ ℝ*)
5 xrltnle 10424 . . . . 5 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
63, 4, 5syl2anc 581 . . . 4 (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
71, 6mpbid 224 . . 3 (𝜑 → ¬ 𝐶𝐵)
87intnand 484 . 2 (𝜑 → ¬ (𝐴𝐶𝐶𝐵))
9 gtnelicc.a . . 3 (𝜑𝐴 ∈ ℝ*)
10 elicc4 12528 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
119, 3, 4, 10syl3anc 1496 . 2 (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
128, 11mtbird 317 1 (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wcel 2166   class class class wbr 4873  (class class class)co 6905  cr 10251  *cxr 10390   < clt 10391  cle 10392  [,]cicc 12466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-xr 10395  df-le 10397  df-icc 12470
This theorem is referenced by:  fourierdlem103  41220
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