Theorem iccgelbd 45826

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

Hypotheses

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

Assertion

Ref	Expression
iccgelbd	⊢ (𝜑 → 𝐴 ≤ 𝐶)

Proof of Theorem iccgelbd

Step	Hyp	Ref	Expression
1		iccgelbd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
2		iccgelbd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
3		iccgelbd.3	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
4		iccgelb 13320	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
5	1, 2, 3, 4	syl3anc 1374	1 ⊢ (𝜑 → 𝐴 ≤ 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2114   class class class wbr 5097  (class class class)co 7358  ℝ^*cxr 11167   ≤ cle 11169  [,]cicc 13266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6447  df-fun 6493  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-xr 11172  df-icc 13270

This theorem is referenced by:  sqrlearg  45836