Theorem iccleubd 45596

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 13301	. 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 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:  sqrlearg  45601