Theorem iccleubd 46775

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 13562	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)
5	1, 2, 3, 4	syl3anc 1399	1 ⊢ (𝜑 → 𝐶 ≤ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   class class class wbr 5106  (class class class)co 7446  ℝ^*cxr 11372   ≤ cle 11374  [,]cicc 13509

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-xr 11377  df-icc 13513

This theorem is used by:  sqrlearg  46780