Theorem iccgelb 13344

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 13331	. . . 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 1144	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
4	3	3impa 1110	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114   class class class wbr 5074  (class class class)co 7356  ℝ^*cxr 11167   ≤ cle 11169  [,]cicc 13290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-pr 5364  ax-un 7678  ax-cnex 11083  ax-resscn 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6443  df-fun 6489  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-xr 11172  df-icc 13294

This theorem is referenced by:  xrge0neqmnf  13394  supicc  13443  ttgcontlem1  28941  xrge0infss  32821  xrge0addgt0  33065  xrge0adddir  33066  esumcst  34195  esumpinfval  34205  oms0  34429  probmeasb  34562  broucube  37963  areaquad  43632  lefldiveq  45713  xadd0ge  45740  xrge0nemnfd  45750  eliccelioc  45939  iccintsng  45941  eliccnelico  45947  eliccelicod  45948  ge0xrre  45949  inficc  45952  iccdificc  45957  iccgelbd  45961  cncfiooiccre  46311  iblspltprt  46389  itgioocnicc  46393  itgspltprt  46395  itgiccshift  46396  fourierdlem1  46524  fourierdlem20  46543  fourierdlem24  46547  fourierdlem25  46548  fourierdlem27  46550  fourierdlem43  46566  fourierdlem44  46567  fourierdlem50  46572  fourierdlem51  46573  fourierdlem52  46574  fourierdlem64  46586  fourierdlem73  46595  fourierdlem76  46598  fourierdlem81  46603  fourierdlem92  46614  fourierdlem102  46624  fourierdlem103  46625  fourierdlem104  46626  fourierdlem114  46636  rrxsnicc  46716  salgencntex  46759  fge0iccico  46786  gsumge0cl  46787  sge0sn  46795  sge0tsms  46796  sge0cl  46797  sge0ge0  46800  sge0fsum  46803  sge0pr  46810  sge0prle  46817  sge0p1  46830  sge0rernmpt  46838  meage0  46891  omessre  46926  omeiunltfirp  46935  carageniuncllem2  46938  omege0  46949  ovnlerp  46978  ovn0lem  46981  hoidmvlelem1  47011  hoidmvlelem2  47012  hoidmvlelem3  47013