Theorem iccgelb 13408

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 13395	. . . 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 1157	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
4	3	3impa 1123	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   ∈ wcel 2144   class class class wbr 5102  (class class class)co 7398  ℝ^*cxr 11217   ≤ cle 11219  [,]cicc 13354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-iota 6479  df-fun 6525  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-xr 11222  df-icc 13358

This theorem is referenced by:  xrge0neqmnf  13458  supicc  13507  ttgcontlem1  29087  xrge0infss  32964  xrge0addgt0  33197  xrge0adddir  33198  esumcst  34362  esumpinfval  34372  oms0  34596  probmeasb  34729  broucube  38158  areaquad  43798  lefldiveq  45876  xadd0ge  45903  xrge0nemnfd  45913  eliccelioc  46102  iccintsng  46104  eliccnelico  46110  eliccelicod  46111  ge0xrre  46112  inficc  46115  iccdificc  46120  iccgelbd  46124  cncfiooiccre  46474  iblspltprt  46552  itgioocnicc  46556  itgspltprt  46558  itgiccshift  46559  fourierdlem1  46687  fourierdlem20  46706  fourierdlem24  46710  fourierdlem25  46711  fourierdlem27  46713  fourierdlem43  46729  fourierdlem44  46730  fourierdlem50  46735  fourierdlem51  46736  fourierdlem52  46737  fourierdlem64  46749  fourierdlem73  46758  fourierdlem76  46761  fourierdlem81  46766  fourierdlem92  46777  fourierdlem102  46787  fourierdlem103  46788  fourierdlem104  46789  fourierdlem114  46799  rrxsnicc  46879  salgencntex  46922  fge0iccico  46949  gsumge0cl  46950  sge0sn  46958  sge0tsms  46959  sge0cl  46960  sge0ge0  46963  sge0fsum  46966  sge0pr  46973  sge0prle  46980  sge0p1  46993  sge0rernmpt  47001  meage0  47054  omessre  47089  omeiunltfirp  47098  carageniuncllem2  47101  omege0  47112  ovnlerp  47141  ovn0lem  47144  hoidmvlelem1  47174  hoidmvlelem2  47175  hoidmvlelem3  47176