 Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  uniin1 Structured version   Visualization version   GIF version

Theorem uniin1 29915
 Description: Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
Assertion
Ref Expression
uniin1 𝑥𝐴 (𝑥𝐵) = ( 𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem uniin1
StepHypRef Expression
1 iunin1 4805 . 2 𝑥𝐴 (𝑥𝐵) = ( 𝑥𝐴 𝑥𝐵)
2 uniiun 4793 . . 3 𝐴 = 𝑥𝐴 𝑥
32ineq1i 4037 . 2 ( 𝐴𝐵) = ( 𝑥𝐴 𝑥𝐵)
41, 3eqtr4i 2852 1 𝑥𝐴 (𝑥𝐵) = ( 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1658   ∩ cin 3797  ∪ cuni 4658  ∪ ciun 4740 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-v 3416  df-in 3805  df-ss 3812  df-uni 4659  df-iun 4742 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator