Theorem iccleub 13427

Description: An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.)

Assertion

Ref	Expression
iccleub	⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)

Proof of Theorem iccleub

Step	Hyp	Ref	Expression
1		elicc1 13415	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2		simp3 1154	. . 3 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ≤ 𝐵)
3	1, 2	biimtrdi 256	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) → 𝐶 ≤ 𝐵))
4	3	3impia 1133	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   ∈ wcel 2149   class class class wbr 5113  (class class class)co 7411  ℝ^*cxr 11241   ≤ cle 11243  [,]cicc 13374

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 5261  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156

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 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  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 11246  df-icc 13378

This theorem is referenced by:  supicc  13527  supiccub  13528  supicclub  13529  oprpiece1res1  25078  ivthlem1  25578  isosctrlem1  26948  ttgcontlem1  29174  broucube  38192  mblfinlem1  38195  ftc1cnnclem  38229  ftc2nc  38240  areaquad  43834  isosctrlem1ALT  45533  lefldiveq  45902  eliccelioc  46128  iccintsng  46130  eliccnelico  46136  eliccelicod  46137  inficc  46141  iccdificc  46146  iccleubd  46155  cncfiooiccre  46500  itgioocnicc  46582  itgspltprt  46584  itgiccshift  46585  fourierdlem1  46713  fourierdlem20  46732  fourierdlem24  46736  fourierdlem25  46737  fourierdlem27  46739  fourierdlem43  46755  fourierdlem44  46756  fourierdlem50  46761  fourierdlem51  46762  fourierdlem52  46763  fourierdlem64  46775  fourierdlem73  46784  fourierdlem76  46787  fourierdlem79  46790  fourierdlem81  46792  fourierdlem92  46803  fourierdlem102  46813  fourierdlem103  46814  fourierdlem104  46815  fourierdlem114  46825  rrxsnicc  46905  salgencntex  46948  sge0p1  47019  hoidmv1lelem3  47198  hoidmvlelem1  47200  hoidmvlelem4  47203