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Theorem iocgtlbd 45424
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 45355 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶)
51, 2, 3, 4syl3anc 1371 1 (𝜑𝐴 < 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2103   class class class wbr 5169  (class class class)co 7445  *cxr 11319   < clt 11320  (,]cioc 13404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766  ax-cnex 11236  ax-resscn 11237
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-iota 6524  df-fun 6574  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-xr 11324  df-ioc 13408
This theorem is referenced by:  xlimpnfvlem1  45692
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