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Theorem xrgtnelicc 45491
Description: A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
xrgtnelicc.1 (𝜑𝐴 ∈ ℝ*)
xrgtnelicc.2 (𝜑𝐵 ∈ ℝ*)
xrgtnelicc.3 (𝜑𝐶 ∈ ℝ*)
xrgtnelicc.4 (𝜑𝐵 < 𝐶)
Assertion
Ref Expression
xrgtnelicc (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Proof of Theorem xrgtnelicc
StepHypRef Expression
1 xrgtnelicc.4 . . . 4 (𝜑𝐵 < 𝐶)
2 xrgtnelicc.2 . . . . 5 (𝜑𝐵 ∈ ℝ*)
3 xrgtnelicc.3 . . . . 5 (𝜑𝐶 ∈ ℝ*)
4 xrltnle 11326 . . . . 5 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
52, 3, 4syl2anc 584 . . . 4 (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
61, 5mpbid 232 . . 3 (𝜑 → ¬ 𝐶𝐵)
76intnand 488 . 2 (𝜑 → ¬ (𝐴𝐶𝐶𝐵))
8 xrgtnelicc.1 . . 3 (𝜑𝐴 ∈ ℝ*)
9 elicc4 13451 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
108, 2, 3, 9syl3anc 1370 . 2 (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
117, 10mtbird 325 1 (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wcel 2106   class class class wbr 5148  (class class class)co 7431  *cxr 11292   < clt 11293  cle 11294  [,]cicc 13387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-xr 11297  df-le 11299  df-icc 13391
This theorem is referenced by:  iccdificc  45492
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