Theorem iocgtlbd 42725

Description: An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

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

Assertion

Ref	Expression
iocgtlbd	⊢ (𝜑 → 𝐴 < 𝐶)

Proof of Theorem iocgtlbd

Step	Hyp	Ref	Expression
1		iocgtlbd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
2		iocgtlbd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
3		iocgtlbd.3	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵))
4		iocgtlb 42656	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶)
5	1, 2, 3, 4	syl3anc 1373	1 ⊢ (𝜑 → 𝐴 < 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2112   class class class wbr 5039  (class class class)co 7191  ℝ^*cxr 10831   < clt 10832  (,]cioc 12901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-resscn 10751

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6316  df-fun 6360  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-xr 10836  df-ioc 12905

This theorem is referenced by:  xlimpnfvlem1  42995