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Theorem invdif 3176
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 2557 . . . . 5 𝑥 ∈ V
2 eldif 2924 . . . . 5 (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐵))
31, 2mpbiran 847 . . . 4 (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥𝐵)
43anbi2i 430 . . 3 ((𝑥𝐴𝑥 ∈ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 elin 3123 . . 3 (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ (𝑥𝐴𝑥 ∈ (V ∖ 𝐵)))
6 eldif 2924 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
74, 5, 63bitr4i 201 . 2 (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
87eqriv 2037 1 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 97   = wceq 1243  wcel 1393  Vcvv 2554  cdif 2911  cin 2913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-dif 2917  df-in 2921
This theorem is referenced by:  indif2  3178  difundir  3187  difindir  3189  difdif2ss  3191  difun1  3194  difdifdirss  3304  nn0supp  8206
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