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Theorem gtnelicc 38569
 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 9968 . . . . 5 (𝜑𝐵 ∈ ℝ*)
4 gtnelicc.c . . . . 5 (𝜑𝐶 ∈ ℝ*)
5 xrltnle 9984 . . . . 5 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
63, 4, 5syl2anc 691 . . . 4 (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
71, 6mpbid 221 . . 3 (𝜑 → ¬ 𝐶𝐵)
87intnand 953 . 2 (𝜑 → ¬ (𝐴𝐶𝐶𝐵))
9 gtnelicc.a . . 3 (𝜑𝐴 ∈ ℝ*)
10 elicc4 12111 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
119, 3, 4, 10syl3anc 1318 . 2 (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
128, 11mtbird 314 1 (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   class class class wbr 4583  (class class class)co 6549  ℝcr 9814  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  [,]cicc 12049 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-xr 9957  df-le 9959  df-icc 12053 This theorem is referenced by:  fourierdlem103  39102
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