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

Theorem difdif 3692
Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
difdif (𝐴 ∖ (𝐵𝐴)) = 𝐴

Proof of Theorem difdif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm4.45im 582 . . 3 (𝑥𝐴 ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)))
2 iman 438 . . . . 5 ((𝑥𝐵𝑥𝐴) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
3 eldif 3544 . . . . 5 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
42, 3xchbinxr 323 . . . 4 ((𝑥𝐵𝑥𝐴) ↔ ¬ 𝑥 ∈ (𝐵𝐴))
54anbi2i 725 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
61, 5bitr2i 263 . 2 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ 𝑥𝐴)
76difeqri 3686 1 (𝐴 ∖ (𝐵𝐴)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1975  cdif 3531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-v 3169  df-dif 3537
This theorem is referenced by:  dif0  3898  undifabs  3991
  Copyright terms: Public domain W3C validator