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Theorem difdif 3728
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 584 . . 3 (𝑥𝐴 ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)))
2 iman 440 . . . . 5 ((𝑥𝐵𝑥𝐴) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
3 eldif 3577 . . . . 5 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
42, 3xchbinxr 325 . . . 4 ((𝑥𝐵𝑥𝐴) ↔ ¬ 𝑥 ∈ (𝐵𝐴))
54anbi2i 729 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
61, 5bitr2i 265 . 2 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ 𝑥𝐴)
76difeqri 3722 1 (𝐴 ∖ (𝐵𝐴)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  cdif 3564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-dif 3570
This theorem is referenced by:  dif0  3941  undifabs  4036
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