Theorem iccgelb 13440

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)) → 𝐴 ≤ 𝐶)

Proof of Theorem iccgelb

Step	Hyp	Ref	Expression
1		elicc1 13428	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2	1	biimpa 476	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
3	2	simp2d 1142	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
4	3	3impa 1109	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2106   class class class wbr 5148  (class class class)co 7431  ℝ^*cxr 11292   ≤ cle 11294  [,]cicc 13387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-xr 11297  df-icc 13391

This theorem is referenced by:  xrge0neqmnf  13489  supicc  13538  ttgcontlem1  28914  xrge0infss  32771  xrge0addgt0  33005  xrge0adddir  33006  esumcst  34044  esumpinfval  34054  oms0  34279  probmeasb  34412  broucube  37641  areaquad  43205  lefldiveq  45243  xadd0ge  45271  xrge0nemnfd  45282  eliccelioc  45474  iccintsng  45476  eliccnelico  45482  eliccelicod  45483  ge0xrre  45484  inficc  45487  iccdificc  45492  iccgelbd  45496  cncfiooiccre  45851  iblspltprt  45929  itgioocnicc  45933  itgspltprt  45935  itgiccshift  45936  fourierdlem1  46064  fourierdlem20  46083  fourierdlem24  46087  fourierdlem25  46088  fourierdlem27  46090  fourierdlem43  46106  fourierdlem44  46107  fourierdlem50  46112  fourierdlem51  46113  fourierdlem52  46114  fourierdlem64  46126  fourierdlem73  46135  fourierdlem76  46138  fourierdlem81  46143  fourierdlem92  46154  fourierdlem102  46164  fourierdlem103  46165  fourierdlem104  46166  fourierdlem114  46176  rrxsnicc  46256  salgencntex  46299  fge0iccico  46326  gsumge0cl  46327  sge0sn  46335  sge0tsms  46336  sge0cl  46337  sge0ge0  46340  sge0fsum  46343  sge0pr  46350  sge0prle  46357  sge0p1  46370  sge0rernmpt  46378  meage0  46431  omessre  46466  omeiunltfirp  46475  carageniuncllem2  46478  omege0  46489  ovnlerp  46518  ovn0lem  46521  hoidmvlelem1  46551  hoidmvlelem2  46552  hoidmvlelem3  46553