Theorem iccleub 13362

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 13350	. . 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 2109   class class class wbr 5107  (class class class)co 7387  ℝ^*cxr 11207   ≤ cle 11209  [,]cicc 13309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-xr 11212  df-icc 13313

This theorem is referenced by:  supicc  13462  supiccub  13463  supicclub  13464  oprpiece1res1  24849  ivthlem1  25352  isosctrlem1  26728  ttgcontlem1  28812  broucube  37648  mblfinlem1  37651  ftc1cnnclem  37685  ftc2nc  37696  areaquad  43205  isosctrlem1ALT  44923  lefldiveq  45290  eliccelioc  45519  iccintsng  45521  eliccnelico  45527  eliccelicod  45528  inficc  45532  iccdificc  45537  iccleubd  45546  cncfiooiccre  45893  itgioocnicc  45975  itgspltprt  45977  itgiccshift  45978  fourierdlem1  46106  fourierdlem20  46125  fourierdlem24  46129  fourierdlem25  46130  fourierdlem27  46132  fourierdlem43  46148  fourierdlem44  46149  fourierdlem50  46154  fourierdlem51  46155  fourierdlem52  46156  fourierdlem64  46168  fourierdlem73  46177  fourierdlem76  46180  fourierdlem79  46183  fourierdlem81  46185  fourierdlem92  46196  fourierdlem102  46206  fourierdlem103  46207  fourierdlem104  46208  fourierdlem114  46218  rrxsnicc  46298  salgencntex  46341  sge0p1  46412  hoidmv1lelem3  46591  hoidmvlelem1  46593  hoidmvlelem4  46596