Theorem iocgtlbd 44274

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 44205	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶)
5	1, 2, 3, 4	syl3anc 1371	1 ⊢ (𝜑 → 𝐴 < 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2106   class class class wbr 5148  (class class class)co 7408  ℝ^*cxr 11246   < clt 11247  (,]cioc 13324

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-xr 11251  df-ioc 13328

This theorem is referenced by:  xlimpnfvlem1  44542