Theorem iccleub 13315

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 13303	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2		simp3 1138	. . 3 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ≤ 𝐵)
3	1, 2	biimtrdi 253	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) → 𝐶 ≤ 𝐵))
4	3	3impia 1117	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2113   class class class wbr 5096  (class class class)co 7356  ℝ^*cxr 11163   ≤ cle 11165  [,]cicc 13262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-xr 11168  df-icc 13266

This theorem is referenced by:  supicc  13415  supiccub  13416  supicclub  13417  oprpiece1res1  24903  ivthlem1  25406  isosctrlem1  26782  ttgcontlem1  28906  broucube  37794  mblfinlem1  37797  ftc1cnnclem  37831  ftc2nc  37842  areaquad  43400  isosctrlem1ALT  45116  lefldiveq  45482  eliccelioc  45709  iccintsng  45711  eliccnelico  45717  eliccelicod  45718  inficc  45722  iccdificc  45727  iccleubd  45736  cncfiooiccre  46081  itgioocnicc  46163  itgspltprt  46165  itgiccshift  46166  fourierdlem1  46294  fourierdlem20  46313  fourierdlem24  46317  fourierdlem25  46318  fourierdlem27  46320  fourierdlem43  46336  fourierdlem44  46337  fourierdlem50  46342  fourierdlem51  46343  fourierdlem52  46344  fourierdlem64  46356  fourierdlem73  46365  fourierdlem76  46368  fourierdlem79  46371  fourierdlem81  46373  fourierdlem92  46384  fourierdlem102  46394  fourierdlem103  46395  fourierdlem104  46396  fourierdlem114  46406  rrxsnicc  46486  salgencntex  46529  sge0p1  46600  hoidmv1lelem3  46779  hoidmvlelem1  46781  hoidmvlelem4  46784