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Theorem difdif 3228
 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 108 . . 3 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) → 𝑥𝐴)
2 pm4.45im 332 . . . 4 (𝑥𝐴 ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)))
3 imanim 678 . . . . . 6 ((𝑥𝐵𝑥𝐴) → ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
4 eldif 3107 . . . . . 6 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
53, 4sylnibr 667 . . . . 5 ((𝑥𝐵𝑥𝐴) → ¬ 𝑥 ∈ (𝐵𝐴))
65anim2i 340 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)) → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
72, 6sylbi 120 . . 3 (𝑥𝐴 → (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
81, 7impbii 125 . 2 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ 𝑥𝐴)
98difeqri 3223 1 (𝐴 ∖ (𝐵𝐴)) = 𝐴
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 2125   ∖ cdif 3095 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-dif 3100 This theorem is referenced by:  dif0  3460
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