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Theorem difdif 3096
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 106 . . 3 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) → 𝑥𝐴)
2 pm4.45im 321 . . . 4 (𝑥𝐴 ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)))
3 imanim 796 . . . . . 6 ((𝑥𝐵𝑥𝐴) → ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
4 eldif 2954 . . . . . 6 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
53, 4sylnibr 612 . . . . 5 ((𝑥𝐵𝑥𝐴) → ¬ 𝑥 ∈ (𝐵𝐴))
65anim2i 328 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)) → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
72, 6sylbi 118 . . 3 (𝑥𝐴 → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
81, 7impbii 121 . 2 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ 𝑥𝐴)
98difeqri 3091 1 (𝐴 ∖ (𝐵𝐴)) = 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101   = wceq 1259  wcel 1409  cdif 2941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947
This theorem is referenced by:  dif0  3321
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