Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inindif Structured version   Visualization version   GIF version

Theorem inindif 30261
 Description: See inundif 4399. (Contributed by Thierry Arnoux, 13-Sep-2017.)
Assertion
Ref Expression
inindif ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅

Proof of Theorem inindif
StepHypRef Expression
1 inss2 4180 . . . 4 (𝐴𝐶) ⊆ 𝐶
21orci 861 . . 3 ((𝐴𝐶) ⊆ 𝐶𝐴𝐶)
3 inss 4189 . . 3 (((𝐴𝐶) ⊆ 𝐶𝐴𝐶) → ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶)
42, 3ax-mp 5 . 2 ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶
5 inssdif0 4301 . 2 (((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
64, 5mpbi 232 1 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 843   = wceq 1537   ∖ cdif 3906   ∩ cin 3908   ⊆ wss 3909  ∅c0 4265 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3472  df-dif 3912  df-in 3916  df-ss 3926  df-nul 4266 This theorem is referenced by:  resf1o  30449  gsummptres  30694  indsumin  31285  measunl  31479  carsgclctun  31583  probdif  31682  hgt750lemd  31923
 Copyright terms: Public domain W3C validator