Theorem iccgelb 13350

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 13337	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2	1	biimpa 478	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
3	2	simp2d 1150	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
4	3	3impa 1116	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   ∈ wcel 2121   class class class wbr 5075  (class class class)co 7360  ℝ^*cxr 11173   ≤ cle 11175  [,]cicc 13296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-xr 11178  df-icc 13300

This theorem is referenced by:  xrge0neqmnf  13400  supicc  13449  ttgcontlem1  28975  xrge0infss  32856  xrge0addgt0  33100  xrge0adddir  33101  esumcst  34259  esumpinfval  34269  oms0  34493  probmeasb  34626  broucube  38036  areaquad  43676  lefldiveq  45754  xadd0ge  45781  xrge0nemnfd  45791  eliccelioc  45980  iccintsng  45982  eliccnelico  45988  eliccelicod  45989  ge0xrre  45990  inficc  45993  iccdificc  45998  iccgelbd  46002  cncfiooiccre  46352  iblspltprt  46430  itgioocnicc  46434  itgspltprt  46436  itgiccshift  46437  fourierdlem1  46565  fourierdlem20  46584  fourierdlem24  46588  fourierdlem25  46589  fourierdlem27  46591  fourierdlem43  46607  fourierdlem44  46608  fourierdlem50  46613  fourierdlem51  46614  fourierdlem52  46615  fourierdlem64  46627  fourierdlem73  46636  fourierdlem76  46639  fourierdlem81  46644  fourierdlem92  46655  fourierdlem102  46665  fourierdlem103  46666  fourierdlem104  46667  fourierdlem114  46677  rrxsnicc  46757  salgencntex  46800  fge0iccico  46827  gsumge0cl  46828  sge0sn  46836  sge0tsms  46837  sge0cl  46838  sge0ge0  46841  sge0fsum  46844  sge0pr  46851  sge0prle  46858  sge0p1  46871  sge0rernmpt  46879  meage0  46932  omessre  46967  omeiunltfirp  46976  carageniuncllem2  46979  omege0  46990  ovnlerp  47019  ovn0lem  47022  hoidmvlelem1  47052  hoidmvlelem2  47053  hoidmvlelem3  47054