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Theorem ltnelicc 38566
Description: A real number smaller than the lower bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
ltnelicc.a (𝜑𝐴 ∈ ℝ)
ltnelicc.b (𝜑𝐵 ∈ ℝ*)
ltnelicc.c (𝜑𝐶 ∈ ℝ*)
ltnelicc.clta (𝜑𝐶 < 𝐴)
Assertion
Ref Expression
ltnelicc (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Proof of Theorem ltnelicc
StepHypRef Expression
1 ltnelicc.clta . . . 4 (𝜑𝐶 < 𝐴)
2 ltnelicc.c . . . . 5 (𝜑𝐶 ∈ ℝ*)
3 ltnelicc.a . . . . . 6 (𝜑𝐴 ∈ ℝ)
43rexrd 9968 . . . . 5 (𝜑𝐴 ∈ ℝ*)
5 xrltnle 9984 . . . . 5 ((𝐶 ∈ ℝ*𝐴 ∈ ℝ*) → (𝐶 < 𝐴 ↔ ¬ 𝐴𝐶))
62, 4, 5syl2anc 691 . . . 4 (𝜑 → (𝐶 < 𝐴 ↔ ¬ 𝐴𝐶))
71, 6mpbid 221 . . 3 (𝜑 → ¬ 𝐴𝐶)
87intnanrd 954 . 2 (𝜑 → ¬ (𝐴𝐶𝐶𝐵))
9 ltnelicc.b . . 3 (𝜑𝐵 ∈ ℝ*)
10 elicc4 12111 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
114, 9, 2, 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  cr 9814  *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:  fourierdlem104  39103
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