Theorem iccleub 13301

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 13289	. . 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 2111   class class class wbr 5089  (class class class)co 7346  ℝ^*cxr 11145   ≤ cle 11147  [,]cicc 13248

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-xr 11150  df-icc 13252

This theorem is referenced by:  supicc  13401  supiccub  13402  supicclub  13403  oprpiece1res1  24876  ivthlem1  25379  isosctrlem1  26755  ttgcontlem1  28863  broucube  37702  mblfinlem1  37705  ftc1cnnclem  37739  ftc2nc  37750  areaquad  43257  isosctrlem1ALT  44974  lefldiveq  45341  eliccelioc  45569  iccintsng  45571  eliccnelico  45577  eliccelicod  45578  inficc  45582  iccdificc  45587  iccleubd  45596  cncfiooiccre  45941  itgioocnicc  46023  itgspltprt  46025  itgiccshift  46026  fourierdlem1  46154  fourierdlem20  46173  fourierdlem24  46177  fourierdlem25  46178  fourierdlem27  46180  fourierdlem43  46196  fourierdlem44  46197  fourierdlem50  46202  fourierdlem51  46203  fourierdlem52  46204  fourierdlem64  46216  fourierdlem73  46225  fourierdlem76  46228  fourierdlem79  46231  fourierdlem81  46233  fourierdlem92  46244  fourierdlem102  46254  fourierdlem103  46255  fourierdlem104  46256  fourierdlem114  46266  rrxsnicc  46346  salgencntex  46389  sge0p1  46460  hoidmv1lelem3  46639  hoidmvlelem1  46641  hoidmvlelem4  46644