Theorem iccleubd 45561

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 13442	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)
5	1, 2, 3, 4	syl3anc 1373	1 ⊢ (𝜑 → 𝐶 ≤ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2108   class class class wbr 5143  (class class class)co 7431  ℝ^*cxr 11294   ≤ cle 11296  [,]cicc 13390

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-xr 11299  df-icc 13394

This theorem is referenced by:  sqrlearg  45566