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Theorem xrgtnelicc 41279
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 10506 . . . . 5 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
52, 3, 4syl2anc 576 . . . 4 (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
61, 5mpbid 224 . . 3 (𝜑 → ¬ 𝐶𝐵)
76intnand 481 . 2 (𝜑 → ¬ (𝐴𝐶𝐶𝐵))
8 xrgtnelicc.1 . . 3 (𝜑𝐴 ∈ ℝ*)
9 elicc4 12617 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
108, 2, 3, 9syl3anc 1352 . 2 (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
117, 10mtbird 317 1 (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wcel 2051   class class class wbr 4925  (class class class)co 6974  *cxr 10471   < clt 10472  cle 10473  [,]cicc 12555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-iota 6149  df-fun 6187  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-xr 10476  df-le 10478  df-icc 12559
This theorem is referenced by:  iccdificc  41280
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