Theorem iccgelb 12781

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 12770	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2	1	biimpa 480	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
3	2	simp2d 1140	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
4	3	3impa 1107	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111   class class class wbr 5030  (class class class)co 7135  ℝ^*cxr 10663   ≤ cle 10665  [,]cicc 12729

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-xr 10668  df-icc 12733

This theorem is referenced by:  xrge0neqmnf  12830  supicc  12879  ttgcontlem1  26679  xrge0infss  30510  xrge0addgt0  30725  xrge0adddir  30726  esumcst  31432  esumpinfval  31442  oms0  31665  probmeasb  31798  broucube  35091  areaquad  40166  lefldiveq  41924  xadd0ge  41952  xrge0nemnfd  41964  eliccelioc  42158  iccintsng  42160  eliccnelico  42166  eliccelicod  42167  ge0xrre  42168  inficc  42171  iccdificc  42176  iccgelbd  42180  cncfiooiccre  42537  iblspltprt  42615  itgioocnicc  42619  itgspltprt  42621  itgiccshift  42622  fourierdlem1  42750  fourierdlem20  42769  fourierdlem24  42773  fourierdlem25  42774  fourierdlem27  42776  fourierdlem43  42792  fourierdlem44  42793  fourierdlem50  42798  fourierdlem51  42799  fourierdlem52  42800  fourierdlem64  42812  fourierdlem73  42821  fourierdlem76  42824  fourierdlem81  42829  fourierdlem92  42840  fourierdlem102  42850  fourierdlem103  42851  fourierdlem104  42852  fourierdlem114  42862  rrxsnicc  42942  salgencntex  42983  fge0iccico  43009  gsumge0cl  43010  sge0sn  43018  sge0tsms  43019  sge0cl  43020  sge0ge0  43023  sge0fsum  43026  sge0pr  43033  sge0prle  43040  sge0p1  43053  sge0rernmpt  43061  meage0  43114  omessre  43149  omeiunltfirp  43158  carageniuncllem2  43161  omege0  43172  ovnlerp  43201  ovn0lem  43204  hoidmvlelem1  43234  hoidmvlelem2  43235  hoidmvlelem3  43236