ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  invdif GIF version

Theorem invdif 3288
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 2663 . . . . 5 𝑥 ∈ V
2 eldif 3050 . . . . 5 (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐵))
31, 2mpbiran 909 . . . 4 (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥𝐵)
43anbi2i 452 . . 3 ((𝑥𝐴𝑥 ∈ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 elin 3229 . . 3 (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ (𝑥𝐴𝑥 ∈ (V ∖ 𝐵)))
6 eldif 3050 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
74, 5, 63bitr4i 211 . 2 (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
87eqriv 2114 1 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1316  wcel 1465  Vcvv 2660  cdif 3038  cin 3040
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047
This theorem is referenced by:  indif2  3290  difundir  3299  difindir  3301  difdif2ss  3303  difun1  3306  difdifdirss  3417  nn0supp  8987
  Copyright terms: Public domain W3C validator