Theorem iccleub 12793

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 12783	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2		simp3 1134	. . 3 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ≤ 𝐵)
3	1, 2	syl6bi 255	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) → 𝐶 ≤ 𝐵))
4	3	3impia 1113	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2114   class class class wbr 5066  (class class class)co 7156  ℝ^*cxr 10674   ≤ cle 10676  [,]cicc 12742

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-xr 10679  df-icc 12746

This theorem is referenced by:  supicc  12887  supiccub  12888  supicclub  12889  oprpiece1res1  23555  ivthlem1  24052  isosctrlem1  25396  ttgcontlem1  26671  broucube  34941  mblfinlem1  34944  ftc1cnnclem  34980  ftc2nc  34991  areaquad  39872  isosctrlem1ALT  41317  lefldiveq  41608  eliccelioc  41846  iccintsng  41848  eliccnelico  41854  eliccelicod  41855  inficc  41859  iccdificc  41864  iccleubd  41873  cncfiooiccre  42227  itgioocnicc  42311  itgspltprt  42313  itgiccshift  42314  fourierdlem1  42442  fourierdlem20  42461  fourierdlem24  42465  fourierdlem25  42466  fourierdlem27  42468  fourierdlem43  42484  fourierdlem44  42485  fourierdlem50  42490  fourierdlem51  42491  fourierdlem52  42492  fourierdlem64  42504  fourierdlem73  42513  fourierdlem76  42516  fourierdlem79  42519  fourierdlem81  42521  fourierdlem92  42532  fourierdlem102  42542  fourierdlem103  42543  fourierdlem104  42544  fourierdlem114  42554  rrxsnicc  42634  salgencntex  42675  sge0p1  42745  hoidmv1lelem3  42924  hoidmvlelem1  42926  hoidmvlelem4  42929