Theorem iccgelb 13305

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 13292	. . . 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 2109   class class class wbr 5092  (class class class)co 7349  ℝ^*cxr 11148   ≤ cle 11150  [,]cicc 13251

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-xr 11153  df-icc 13255

This theorem is referenced by:  xrge0neqmnf  13355  supicc  13404  ttgcontlem1  28834  xrge0infss  32712  xrge0addgt0  32980  xrge0adddir  32981  esumcst  34046  esumpinfval  34056  oms0  34281  probmeasb  34414  broucube  37654  areaquad  43209  lefldiveq  45294  xadd0ge  45321  xrge0nemnfd  45332  eliccelioc  45522  iccintsng  45524  eliccnelico  45530  eliccelicod  45531  ge0xrre  45532  inficc  45535  iccdificc  45540  iccgelbd  45544  cncfiooiccre  45896  iblspltprt  45974  itgioocnicc  45978  itgspltprt  45980  itgiccshift  45981  fourierdlem1  46109  fourierdlem20  46128  fourierdlem24  46132  fourierdlem25  46133  fourierdlem27  46135  fourierdlem43  46151  fourierdlem44  46152  fourierdlem50  46157  fourierdlem51  46158  fourierdlem52  46159  fourierdlem64  46171  fourierdlem73  46180  fourierdlem76  46183  fourierdlem81  46188  fourierdlem92  46199  fourierdlem102  46209  fourierdlem103  46210  fourierdlem104  46211  fourierdlem114  46221  rrxsnicc  46301  salgencntex  46344  fge0iccico  46371  gsumge0cl  46372  sge0sn  46380  sge0tsms  46381  sge0cl  46382  sge0ge0  46385  sge0fsum  46388  sge0pr  46395  sge0prle  46402  sge0p1  46415  sge0rernmpt  46423  meage0  46476  omessre  46511  omeiunltfirp  46520  carageniuncllem2  46523  omege0  46534  ovnlerp  46563  ovn0lem  46566  hoidmvlelem1  46596  hoidmvlelem2  46597  hoidmvlelem3  46598