Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iocgtlbd Structured version   Visualization version   GIF version

Theorem iocgtlbd 43816
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 43747 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶)
51, 2, 3, 4syl3anc 1372 1 (𝜑𝐴 < 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107   class class class wbr 5106  (class class class)co 7358  *cxr 11189   < clt 11190  (,]cioc 13266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-xr 11194  df-ioc 13270
This theorem is referenced by:  xlimpnfvlem1  44084
  Copyright terms: Public domain W3C validator