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Theorem lbioc 39548
Description: An left open right closed interval doesn't contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
lbioc ¬ 𝐴 ∈ (𝐴(,]𝐵)

Proof of Theorem lbioc
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioc 12177 . . . . 5 (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧𝑦)})
21elixx3g 12185 . . . 4 (𝐴 ∈ (𝐴(,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 ∈ ℝ*) ∧ (𝐴 < 𝐴𝐴𝐵)))
32biimpi 206 . . 3 (𝐴 ∈ (𝐴(,]𝐵) → ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 ∈ ℝ*) ∧ (𝐴 < 𝐴𝐴𝐵)))
43simprld 795 . 2 (𝐴 ∈ (𝐴(,]𝐵) → 𝐴 < 𝐴)
51elmpt2cl1 6874 . . 3 (𝐴 ∈ (𝐴(,]𝐵) → 𝐴 ∈ ℝ*)
6 xrltnr 11950 . . 3 (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
75, 6syl 17 . 2 (𝐴 ∈ (𝐴(,]𝐵) → ¬ 𝐴 < 𝐴)
84, 7pm2.65i 185 1 ¬ 𝐴 ∈ (𝐴(,]𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384  w3a 1037  wcel 1989  {crab 2915   class class class wbr 4651  (class class class)co 6647  *cxr 10070   < clt 10071  cle 10072  (,]cioc 12173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-pre-lttri 10007  ax-pre-lttrn 10008
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-po 5033  df-so 5034  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-ioc 12177
This theorem is referenced by:  fouriersw  40217
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