Theorem iccgelb 13429

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 13416	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2	1	biimpa 481	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
3	2	simp2d 1159	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
4	3	3impa 1125	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   ∈ wcel 2149   class class class wbr 5111  (class class class)co 7411  ℝ^*cxr 11242   ≤ cle 11244  [,]cicc 13375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-sbc 3752  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-xr 11247  df-icc 13379

This theorem is referenced by:  xrge0neqmnf  13479  supicc  13528  ttgcontlem1  29175  xrge0infss  33046  xrge0addgt0  33278  xrge0adddir  33279  esumcst  34398  esumpinfval  34408  oms0  34632  probmeasb  34765  broucube  38228  areaquad  43870  lefldiveq  45938  xadd0ge  45965  xrge0nemnfd  45975  eliccelioc  46164  iccintsng  46166  eliccnelico  46172  eliccelicod  46173  ge0xrre  46174  inficc  46177  iccdificc  46182  iccgelbd  46186  cncfiooiccre  46536  iblspltprt  46614  itgioocnicc  46618  itgspltprt  46620  itgiccshift  46621  fourierdlem1  46749  fourierdlem20  46768  fourierdlem24  46772  fourierdlem25  46773  fourierdlem27  46775  fourierdlem43  46791  fourierdlem44  46792  fourierdlem50  46797  fourierdlem51  46798  fourierdlem52  46799  fourierdlem64  46811  fourierdlem73  46820  fourierdlem76  46823  fourierdlem81  46828  fourierdlem92  46839  fourierdlem102  46849  fourierdlem103  46850  fourierdlem104  46851  fourierdlem114  46861  rrxsnicc  46941  salgencntex  46984  fge0iccico  47011  gsumge0cl  47012  sge0sn  47020  sge0tsms  47021  sge0cl  47022  sge0ge0  47025  sge0fsum  47028  sge0pr  47035  sge0prle  47042  sge0p1  47055  sge0rernmpt  47063  meage0  47116  omessre  47151  omeiunltfirp  47160  carageniuncllem2  47163  omege0  47174  ovnlerp  47203  ovn0lem  47206  hoidmvlelem1  47236  hoidmvlelem2  47237  hoidmvlelem3  47238