Theorem iccgelb 13435

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 13422	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2	1	biimpa 481	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
3	2	simp2d 1160	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
4	3	3impa 1126	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   ∈ wcel 2142   class class class wbr 5108  (class class class)co 7412  ℝ^*cxr 11248   ≤ cle 11250  [,]cicc 13381

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-xr 11253  df-icc 13385

This theorem is used by:  xrge0neqmnf  13485  supicc  13534  ttgcontlem1  29245  xrge0infss  33116  xrge0addgt0  33346  xrge0adddir  33347  esumcst  34462  esumpinfval  34472  oms0  34696  probmeasb  34829  broucube  38333  areaquad  43971  lefldiveq  46039  xadd0ge  46066  xrge0nemnfd  46076  eliccelioc  46265  iccintsng  46267  eliccnelico  46273  eliccelicod  46274  ge0xrre  46275  inficc  46278  iccdificc  46283  iccgelbd  46287  cncfiooiccre  46637  iblspltprt  46715  itgioocnicc  46719  itgspltprt  46721  itgiccshift  46722  fourierdlem1  46850  fourierdlem20  46869  fourierdlem24  46873  fourierdlem25  46874  fourierdlem27  46876  fourierdlem43  46892  fourierdlem44  46893  fourierdlem50  46898  fourierdlem51  46899  fourierdlem52  46900  fourierdlem64  46912  fourierdlem73  46921  fourierdlem76  46924  fourierdlem81  46929  fourierdlem92  46940  fourierdlem102  46950  fourierdlem103  46951  fourierdlem104  46952  fourierdlem114  46962  rrxsnicc  47042  salgencntex  47085  fge0iccico  47112  gsumge0cl  47113  sge0sn  47121  sge0tsms  47122  sge0cl  47123  sge0ge0  47126  sge0fsum  47129  sge0pr  47136  sge0prle  47143  sge0p1  47156  sge0rernmpt  47164  meage0  47217  omessre  47252  omeiunltfirp  47261  carageniuncllem2  47264  omege0  47275  ovnlerp  47304  ovn0lem  47307  hoidmvlelem1  47337  hoidmvlelem2  47338  hoidmvlelem3  47339