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Theorem iccgelb 12423
Description: An element of a closed interval is more than or equal to its lower bound. (Contributed by Thierry Arnoux, 23-Dec-2016.)
Assertion
Ref Expression
iccgelb ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐴𝐶)

Proof of Theorem iccgelb
StepHypRef Expression
1 elicc1 12412 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵)))
21biimpa 502 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵))
32simp2d 1138 . 2 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴𝐶)
433impa 1101 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wcel 2139   class class class wbr 4804  (class class class)co 6813  *cxr 10265  cle 10267  [,]cicc 12371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-xr 10270  df-icc 12375
This theorem is referenced by:  supicc  12513  ttgcontlem1  25964  xrge0infss  29834  xrge0addgt0  30000  xrge0adddir  30001  esumcst  30434  esumpinfval  30444  oms0  30668  probmeasb  30801  broucube  33756  areaquad  38304  lefldiveq  40004  xadd0ge  40034  xrge0nemnfd  40046  eliccelioc  40250  iccintsng  40252  ge0nemnf2  40258  eliccnelico  40259  eliccelicod  40260  ge0xrre  40261  inficc  40264  iccdificc  40269  iccgelbd  40273  cncfiooiccre  40611  iblspltprt  40692  itgioocnicc  40696  itgspltprt  40698  itgiccshift  40699  fourierdlem1  40828  fourierdlem20  40847  fourierdlem24  40851  fourierdlem25  40852  fourierdlem27  40854  fourierdlem43  40870  fourierdlem44  40871  fourierdlem50  40876  fourierdlem51  40877  fourierdlem52  40878  fourierdlem64  40890  fourierdlem73  40899  fourierdlem76  40902  fourierdlem81  40907  fourierdlem92  40918  fourierdlem102  40928  fourierdlem103  40929  fourierdlem104  40930  fourierdlem114  40940  rrxsnicc  41023  salgencntex  41064  fge0iccico  41090  gsumge0cl  41091  sge0sn  41099  sge0tsms  41100  sge0cl  41101  sge0ge0  41104  sge0fsum  41107  sge0pr  41114  sge0prle  41121  sge0p1  41134  sge0rernmpt  41142  meage0  41195  omessre  41230  omeiunltfirp  41239  carageniuncllem2  41242  omege0  41253  ovnlerp  41282  ovn0lem  41285  hoidmvlelem1  41315  hoidmvlelem2  41316  hoidmvlelem3  41317
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