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Theorem gtnelicc 44932
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 11304 . . . . 5 (𝜑𝐵 ∈ ℝ*)
4 gtnelicc.c . . . . 5 (𝜑𝐶 ∈ ℝ*)
5 xrltnle 11321 . . . . 5 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
63, 4, 5syl2anc 582 . . . 4 (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
71, 6mpbid 231 . . 3 (𝜑 → ¬ 𝐶𝐵)
87intnand 487 . 2 (𝜑 → ¬ (𝐴𝐶𝐶𝐵))
9 gtnelicc.a . . 3 (𝜑𝐴 ∈ ℝ*)
10 elicc4 13433 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
119, 3, 4, 10syl3anc 1368 . 2 (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
128, 11mtbird 324 1 (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wcel 2098   class class class wbr 5152  (class class class)co 7426  cr 11147  *cxr 11287   < clt 11288  cle 11289  [,]cicc 13369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-xr 11292  df-le 11294  df-icc 13373
This theorem is referenced by:  fourierdlem103  45644
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