Theorem iccleub 13424

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 13413	. . 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 2107   class class class wbr 5123  (class class class)co 7413  ℝ^*cxr 11276   ≤ cle 11278  [,]cicc 13372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-xr 11281  df-icc 13376

This theorem is referenced by:  supicc  13523  supiccub  13524  supicclub  13525  oprpiece1res1  24918  ivthlem1  25422  isosctrlem1  26797  ttgcontlem1  28830  broucube  37620  mblfinlem1  37623  ftc1cnnclem  37657  ftc2nc  37668  areaquad  43191  isosctrlem1ALT  44911  lefldiveq  45261  eliccelioc  45491  iccintsng  45493  eliccnelico  45499  eliccelicod  45500  inficc  45504  iccdificc  45509  iccleubd  45518  cncfiooiccre  45867  itgioocnicc  45949  itgspltprt  45951  itgiccshift  45952  fourierdlem1  46080  fourierdlem20  46099  fourierdlem24  46103  fourierdlem25  46104  fourierdlem27  46106  fourierdlem43  46122  fourierdlem44  46123  fourierdlem50  46128  fourierdlem51  46129  fourierdlem52  46130  fourierdlem64  46142  fourierdlem73  46151  fourierdlem76  46154  fourierdlem79  46157  fourierdlem81  46159  fourierdlem92  46170  fourierdlem102  46180  fourierdlem103  46181  fourierdlem104  46182  fourierdlem114  46192  rrxsnicc  46272  salgencntex  46315  sge0p1  46386  hoidmv1lelem3  46565  hoidmvlelem1  46567  hoidmvlelem4  46570