MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  undif3 Structured version   Visualization version   GIF version

Theorem undif3 4262
Description: An equality involving class union and class difference. The first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 17-Apr-2012.) (Proof shortened by JJ, 13-Jul-2021.)
Assertion
Ref Expression
undif3 (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶𝐴))

Proof of Theorem undif3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elun 4122 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 pm4.53 979 . . . . 5 (¬ (𝑥𝐶 ∧ ¬ 𝑥𝐴) ↔ (¬ 𝑥𝐶𝑥𝐴))
3 eldif 3943 . . . . 5 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
42, 3xchnxbir 334 . . . 4 𝑥 ∈ (𝐶𝐴) ↔ (¬ 𝑥𝐶𝑥𝐴))
51, 4anbi12i 626 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
6 eldif 3943 . . 3 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
7 elun 4122 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
8 eldif 3943 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
98orbi2i 906 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
10 ordi 999 . . . . 5 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)))
11 orcom 864 . . . . . 6 ((𝑥𝐴 ∨ ¬ 𝑥𝐶) ↔ (¬ 𝑥𝐶𝑥𝐴))
1211anbi2i 622 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
1310, 12bitri 276 . . . 4 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
147, 9, 133bitri 298 . . 3 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
155, 6, 143bitr4ri 305 . 2 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)))
1615eqriv 2815 1 (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wo 841   = wceq 1528  wcel 2105  cdif 3930  cun 3931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-dif 3936  df-un 3938
This theorem is referenced by:  undifabs  4422  llycmpkgen2  22086  hgt750lemb  31826
  Copyright terms: Public domain W3C validator