Theorem iccleubd 43040

Description: An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
iccleubd.1	⊢ (𝜑 → 𝐴 ∈ ℝ*)
iccleubd.2	⊢ (𝜑 → 𝐵 ∈ ℝ*)
iccleubd.3	⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))

Assertion

Ref	Expression
iccleubd	⊢ (𝜑 → 𝐶 ≤ 𝐵)

Proof of Theorem iccleubd

Step	Hyp	Ref	Expression
1		iccleubd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
2		iccleubd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
3		iccleubd.3	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
4		iccleub 13116	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)
5	1, 2, 3, 4	syl3anc 1369	1 ⊢ (𝜑 → 𝐶 ≤ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2109   class class class wbr 5078  (class class class)co 7268  ℝ^*cxr 10992   ≤ cle 10994  [,]cicc 13064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-iota 6388  df-fun 6432  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-xr 10997  df-icc 13068

This theorem is referenced by:  sqrlearg  43045