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Theorem iocgtlbd 42201
 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 (𝜑𝐴 ∈ ℝ*)
iocgtlbd.2 (𝜑𝐵 ∈ ℝ*)
iocgtlbd.3 (𝜑𝐶 ∈ (𝐴(,]𝐵))
Assertion
Ref Expression
iocgtlbd (𝜑𝐴 < 𝐶)

Proof of Theorem iocgtlbd
StepHypRef Expression
1 iocgtlbd.1 . 2 (𝜑𝐴 ∈ ℝ*)
2 iocgtlbd.2 . 2 (𝜑𝐵 ∈ ℝ*)
3 iocgtlbd.3 . 2 (𝜑𝐶 ∈ (𝐴(,]𝐵))
4 iocgtlb 42132 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶)
51, 2, 3, 4syl3anc 1368 1 (𝜑𝐴 < 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112   class class class wbr 5033  (class class class)co 7139  ℝ*cxr 10667   < clt 10668  (,]cioc 12731 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-xr 10672  df-ioc 12735 This theorem is referenced by:  xlimpnfvlem1  42471
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