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Theorem invdif 3322
Description: Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
invdif (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)

Proof of Theorem invdif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 2692 . . . . 5 𝑥 ∈ V
2 eldif 3084 . . . . 5 (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐵))
31, 2mpbiran 925 . . . 4 (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥𝐵)
43anbi2i 453 . . 3 ((𝑥𝐴𝑥 ∈ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 elin 3263 . . 3 (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ (𝑥𝐴𝑥 ∈ (V ∖ 𝐵)))
6 eldif 3084 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
74, 5, 63bitr4i 211 . 2 (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
87eqriv 2137 1 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1332  wcel 1481  Vcvv 2689  cdif 3072  cin 3074
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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3077  df-in 3081
This theorem is referenced by:  indif2  3324  difundir  3333  difindir  3335  difdif2ss  3337  difun1  3340  difdifdirss  3451  nn0supp  9052
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