Theorem iccgelb 13302

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 13289	. . . 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 2111   class class class wbr 5089  (class class class)co 7346  ℝ^*cxr 11145   ≤ cle 11147  [,]cicc 13248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-xr 11150  df-icc 13252

This theorem is referenced by:  xrge0neqmnf  13352  supicc  13401  ttgcontlem1  28863  xrge0infss  32743  xrge0addgt0  32998  xrge0adddir  32999  esumcst  34076  esumpinfval  34086  oms0  34310  probmeasb  34443  broucube  37693  areaquad  43308  lefldiveq  45392  xadd0ge  45419  xrge0nemnfd  45430  eliccelioc  45620  iccintsng  45622  eliccnelico  45628  eliccelicod  45629  ge0xrre  45630  inficc  45633  iccdificc  45638  iccgelbd  45642  cncfiooiccre  45992  iblspltprt  46070  itgioocnicc  46074  itgspltprt  46076  itgiccshift  46077  fourierdlem1  46205  fourierdlem20  46224  fourierdlem24  46228  fourierdlem25  46229  fourierdlem27  46231  fourierdlem43  46247  fourierdlem44  46248  fourierdlem50  46253  fourierdlem51  46254  fourierdlem52  46255  fourierdlem64  46267  fourierdlem73  46276  fourierdlem76  46279  fourierdlem81  46284  fourierdlem92  46295  fourierdlem102  46305  fourierdlem103  46306  fourierdlem104  46307  fourierdlem114  46317  rrxsnicc  46397  salgencntex  46440  fge0iccico  46467  gsumge0cl  46468  sge0sn  46476  sge0tsms  46477  sge0cl  46478  sge0ge0  46481  sge0fsum  46484  sge0pr  46491  sge0prle  46498  sge0p1  46511  sge0rernmpt  46519  meage0  46572  omessre  46607  omeiunltfirp  46616  carageniuncllem2  46619  omege0  46630  ovnlerp  46659  ovn0lem  46662  hoidmvlelem1  46692  hoidmvlelem2  46693  hoidmvlelem3  46694