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Theorem ltnelicc 42575
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 10769 . . . . 5 (𝜑𝐴 ∈ ℝ*)
5 xrltnle 10786 . . . . 5 ((𝐶 ∈ ℝ*𝐴 ∈ ℝ*) → (𝐶 < 𝐴 ↔ ¬ 𝐴𝐶))
62, 4, 5syl2anc 587 . . . 4 (𝜑 → (𝐶 < 𝐴 ↔ ¬ 𝐴𝐶))
71, 6mpbid 235 . . 3 (𝜑 → ¬ 𝐴𝐶)
87intnanrd 493 . 2 (𝜑 → ¬ (𝐴𝐶𝐶𝐵))
9 ltnelicc.b . . 3 (𝜑𝐵 ∈ ℝ*)
10 elicc4 12888 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
114, 9, 2, 10syl3anc 1372 . 2 (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
128, 11mtbird 328 1 (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2114   class class class wbr 5030  (class class class)co 7170  cr 10614  *cxr 10752   < clt 10753  cle 10754  [,]cicc 12824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-xr 10757  df-le 10759  df-icc 12828
This theorem is referenced by:  fourierdlem104  43293
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