Theorem iccleub 12786

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 12776	. . 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 2110   class class class wbr 5059  (class class class)co 7150  ℝ^*cxr 10668   ≤ cle 10670  [,]cicc 12735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-iota 6309  df-fun 6352  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-xr 10673  df-icc 12739

This theorem is referenced by:  supicc  12880  supiccub  12881  supicclub  12882  oprpiece1res1  23549  ivthlem1  24046  isosctrlem1  25390  ttgcontlem1  26665  broucube  34920  mblfinlem1  34923  ftc1cnnclem  34959  ftc2nc  34970  areaquad  39816  isosctrlem1ALT  41261  lefldiveq  41551  eliccelioc  41789  iccintsng  41791  eliccnelico  41797  eliccelicod  41798  inficc  41802  iccdificc  41807  iccleubd  41816  cncfiooiccre  42170  itgioocnicc  42254  itgspltprt  42256  itgiccshift  42257  fourierdlem1  42386  fourierdlem20  42405  fourierdlem24  42409  fourierdlem25  42410  fourierdlem27  42412  fourierdlem43  42428  fourierdlem44  42429  fourierdlem50  42434  fourierdlem51  42435  fourierdlem52  42436  fourierdlem64  42448  fourierdlem73  42457  fourierdlem76  42460  fourierdlem79  42463  fourierdlem81  42465  fourierdlem92  42476  fourierdlem102  42486  fourierdlem103  42487  fourierdlem104  42488  fourierdlem114  42498  rrxsnicc  42578  salgencntex  42619  sge0p1  42689  hoidmv1lelem3  42868  hoidmvlelem1  42870  hoidmvlelem4  42873