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Theorem difdif 3114
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 simpl 107 . . 3 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) → 𝑥𝐴)
2 pm4.45im 327 . . . 4 (𝑥𝐴 ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)))
3 imanim 821 . . . . . 6 ((𝑥𝐵𝑥𝐴) → ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
4 eldif 2997 . . . . . 6 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
53, 4sylnibr 635 . . . . 5 ((𝑥𝐵𝑥𝐴) → ¬ 𝑥 ∈ (𝐵𝐴))
65anim2i 334 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)) → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
72, 6sylbi 119 . . 3 (𝑥𝐴 → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
81, 7impbii 124 . 2 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ 𝑥𝐴)
98difeqri 3109 1 (𝐴 ∖ (𝐵𝐴)) = 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102   = wceq 1287  wcel 1436  cdif 2985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-dif 2990
This theorem is referenced by:  dif0  3341
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