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Theorem xrgtnelicc 44562
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 11288 . . . . 5 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
52, 3, 4syl2anc 583 . . . 4 (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
61, 5mpbid 231 . . 3 (𝜑 → ¬ 𝐶𝐵)
76intnand 488 . 2 (𝜑 → ¬ (𝐴𝐶𝐶𝐵))
8 xrgtnelicc.1 . . 3 (𝜑𝐴 ∈ ℝ*)
9 elicc4 13398 . . 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 205  wa 395  wcel 2105   class class class wbr 5148  (class class class)co 7412  *cxr 11254   < clt 11255  cle 11256  [,]cicc 13334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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 7415  df-oprab 7416  df-mpo 7417  df-xr 11259  df-le 11261  df-icc 13338
This theorem is referenced by:  iccdificc  44563
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