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Theorem ltnelicc 40222
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 10281 . . . . 5 (𝜑𝐴 ∈ ℝ*)
5 xrltnle 10297 . . . . 5 ((𝐶 ∈ ℝ*𝐴 ∈ ℝ*) → (𝐶 < 𝐴 ↔ ¬ 𝐴𝐶))
62, 4, 5syl2anc 696 . . . 4 (𝜑 → (𝐶 < 𝐴 ↔ ¬ 𝐴𝐶))
71, 6mpbid 222 . . 3 (𝜑 → ¬ 𝐴𝐶)
87intnanrd 1001 . 2 (𝜑 → ¬ (𝐴𝐶𝐶𝐵))
9 ltnelicc.b . . 3 (𝜑𝐵 ∈ ℝ*)
10 elicc4 12433 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
114, 9, 2, 10syl3anc 1477 . 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 196  wa 383  wcel 2139   class class class wbr 4804  (class class class)co 6813  cr 10127  *cxr 10265   < clt 10266  cle 10267  [,]cicc 12371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-xr 10270  df-le 10272  df-icc 12375
This theorem is referenced by:  fourierdlem104  40930
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