Theorem iccgelb 13370

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 13357	. . . 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 5110  (class class class)co 7390  ℝ^*cxr 11214   ≤ cle 11216  [,]cicc 13316

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-xr 11219  df-icc 13320

This theorem is referenced by:  xrge0neqmnf  13420  supicc  13469  ttgcontlem1  28819  xrge0infss  32690  xrge0addgt0  32965  xrge0adddir  32966  esumcst  34060  esumpinfval  34070  oms0  34295  probmeasb  34428  broucube  37655  areaquad  43212  lefldiveq  45297  xadd0ge  45324  xrge0nemnfd  45335  eliccelioc  45526  iccintsng  45528  eliccnelico  45534  eliccelicod  45535  ge0xrre  45536  inficc  45539  iccdificc  45544  iccgelbd  45548  cncfiooiccre  45900  iblspltprt  45978  itgioocnicc  45982  itgspltprt  45984  itgiccshift  45985  fourierdlem1  46113  fourierdlem20  46132  fourierdlem24  46136  fourierdlem25  46137  fourierdlem27  46139  fourierdlem43  46155  fourierdlem44  46156  fourierdlem50  46161  fourierdlem51  46162  fourierdlem52  46163  fourierdlem64  46175  fourierdlem73  46184  fourierdlem76  46187  fourierdlem81  46192  fourierdlem92  46203  fourierdlem102  46213  fourierdlem103  46214  fourierdlem104  46215  fourierdlem114  46225  rrxsnicc  46305  salgencntex  46348  fge0iccico  46375  gsumge0cl  46376  sge0sn  46384  sge0tsms  46385  sge0cl  46386  sge0ge0  46389  sge0fsum  46392  sge0pr  46399  sge0prle  46406  sge0p1  46419  sge0rernmpt  46427  meage0  46480  omessre  46515  omeiunltfirp  46524  carageniuncllem2  46527  omege0  46538  ovnlerp  46567  ovn0lem  46570  hoidmvlelem1  46600  hoidmvlelem2  46601  hoidmvlelem3  46602