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Theorem invdif 3282
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 2658 . . . . 5 𝑥 ∈ V
2 eldif 3044 . . . . 5 (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐵))
31, 2mpbiran 905 . . . 4 (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥𝐵)
43anbi2i 450 . . 3 ((𝑥𝐴𝑥 ∈ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 elin 3223 . . 3 (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ (𝑥𝐴𝑥 ∈ (V ∖ 𝐵)))
6 eldif 3044 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
74, 5, 63bitr4i 211 . 2 (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
87eqriv 2110 1 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1312  wcel 1461  Vcvv 2655  cdif 3032  cin 3034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-dif 3037  df-in 3041
This theorem is referenced by:  indif2  3284  difundir  3293  difindir  3295  difdif2ss  3297  difun1  3300  difdifdirss  3411  nn0supp  8927
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