Theorem iccgelb 13463

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 13451	. . . 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 1143	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
4	3	3impa 1110	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108   class class class wbr 5166  (class class class)co 7448  ℝ^*cxr 11323   ≤ cle 11325  [,]cicc 13410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-xr 11328  df-icc 13414

This theorem is referenced by:  xrge0neqmnf  13512  supicc  13561  ttgcontlem1  28917  xrge0infss  32767  xrge0addgt0  33003  xrge0adddir  33004  esumcst  34027  esumpinfval  34037  oms0  34262  probmeasb  34395  broucube  37614  areaquad  43177  lefldiveq  45207  xadd0ge  45235  xrge0nemnfd  45247  eliccelioc  45439  iccintsng  45441  eliccnelico  45447  eliccelicod  45448  ge0xrre  45449  inficc  45452  iccdificc  45457  iccgelbd  45461  cncfiooiccre  45816  iblspltprt  45894  itgioocnicc  45898  itgspltprt  45900  itgiccshift  45901  fourierdlem1  46029  fourierdlem20  46048  fourierdlem24  46052  fourierdlem25  46053  fourierdlem27  46055  fourierdlem43  46071  fourierdlem44  46072  fourierdlem50  46077  fourierdlem51  46078  fourierdlem52  46079  fourierdlem64  46091  fourierdlem73  46100  fourierdlem76  46103  fourierdlem81  46108  fourierdlem92  46119  fourierdlem102  46129  fourierdlem103  46130  fourierdlem104  46131  fourierdlem114  46141  rrxsnicc  46221  salgencntex  46264  fge0iccico  46291  gsumge0cl  46292  sge0sn  46300  sge0tsms  46301  sge0cl  46302  sge0ge0  46305  sge0fsum  46308  sge0pr  46315  sge0prle  46322  sge0p1  46335  sge0rernmpt  46343  meage0  46396  omessre  46431  omeiunltfirp  46440  carageniuncllem2  46443  omege0  46454  ovnlerp  46483  ovn0lem  46486  hoidmvlelem1  46516  hoidmvlelem2  46517  hoidmvlelem3  46518