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Theorem ltnelicc 44289
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 11266 . . . . 5 (𝜑𝐴 ∈ ℝ*)
5 xrltnle 11283 . . . . 5 ((𝐶 ∈ ℝ*𝐴 ∈ ℝ*) → (𝐶 < 𝐴 ↔ ¬ 𝐴𝐶))
62, 4, 5syl2anc 584 . . . 4 (𝜑 → (𝐶 < 𝐴 ↔ ¬ 𝐴𝐶))
71, 6mpbid 231 . . 3 (𝜑 → ¬ 𝐴𝐶)
87intnanrd 490 . 2 (𝜑 → ¬ (𝐴𝐶𝐶𝐵))
9 ltnelicc.b . . 3 (𝜑𝐵 ∈ ℝ*)
10 elicc4 13393 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
114, 9, 2, 10syl3anc 1371 . 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 396  wcel 2106   class class class wbr 5148  (class class class)co 7411  cr 11111  *cxr 11249   < clt 11250  cle 11251  [,]cicc 13329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-xr 11254  df-le 11256  df-icc 13333
This theorem is referenced by:  fourierdlem104  45005
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