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Theorem xrgtnelicc 42168
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 10701 . . . . 5 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
52, 3, 4syl2anc 587 . . . 4 (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
61, 5mpbid 235 . . 3 (𝜑 → ¬ 𝐶𝐵)
76intnand 492 . 2 (𝜑 → ¬ (𝐴𝐶𝐶𝐵))
8 xrgtnelicc.1 . . 3 (𝜑𝐴 ∈ ℝ*)
9 elicc4 12796 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
108, 2, 3, 9syl3anc 1368 . 2 (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))
117, 10mtbird 328 1 (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2112   class class class wbr 5033  (class class class)co 7139  *cxr 10667   < clt 10668  cle 10669  [,]cicc 12733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-xr 10672  df-le 10674  df-icc 12737
This theorem is referenced by:  iccdificc  42169
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