Theorem iccgelb 13419

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 13406	. . . 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 1109	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2108   class class class wbr 5119  (class class class)co 7405  ℝ^*cxr 11268   ≤ cle 11270  [,]cicc 13365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-xr 11273  df-icc 13369

This theorem is referenced by:  xrge0neqmnf  13469  supicc  13518  ttgcontlem1  28864  xrge0infss  32737  xrge0addgt0  33012  xrge0adddir  33013  esumcst  34094  esumpinfval  34104  oms0  34329  probmeasb  34462  broucube  37678  areaquad  43240  lefldiveq  45321  xadd0ge  45348  xrge0nemnfd  45359  eliccelioc  45550  iccintsng  45552  eliccnelico  45558  eliccelicod  45559  ge0xrre  45560  inficc  45563  iccdificc  45568  iccgelbd  45572  cncfiooiccre  45924  iblspltprt  46002  itgioocnicc  46006  itgspltprt  46008  itgiccshift  46009  fourierdlem1  46137  fourierdlem20  46156  fourierdlem24  46160  fourierdlem25  46161  fourierdlem27  46163  fourierdlem43  46179  fourierdlem44  46180  fourierdlem50  46185  fourierdlem51  46186  fourierdlem52  46187  fourierdlem64  46199  fourierdlem73  46208  fourierdlem76  46211  fourierdlem81  46216  fourierdlem92  46227  fourierdlem102  46237  fourierdlem103  46238  fourierdlem104  46239  fourierdlem114  46249  rrxsnicc  46329  salgencntex  46372  fge0iccico  46399  gsumge0cl  46400  sge0sn  46408  sge0tsms  46409  sge0cl  46410  sge0ge0  46413  sge0fsum  46416  sge0pr  46423  sge0prle  46430  sge0p1  46443  sge0rernmpt  46451  meage0  46504  omessre  46539  omeiunltfirp  46548  carageniuncllem2  46551  omege0  46562  ovnlerp  46591  ovn0lem  46594  hoidmvlelem1  46624  hoidmvlelem2  46625  hoidmvlelem3  46626