Theorem iccleubd 46386

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 13458	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)
5	1, 2, 3, 4	syl3anc 1398	1 ⊢ (𝜑 → 𝐶 ≤ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   class class class wbr 5107  (class class class)co 7417  ℝ^*cxr 11270   ≤ cle 11272  [,]cicc 13405

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-xr 11275  df-icc 13409

This theorem is used by:  sqrlearg  46391