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

Theorem invdif 3379
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 2742 . . . . 5 𝑥 ∈ V
2 eldif 3140 . . . . 5 (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐵))
31, 2mpbiran 940 . . . 4 (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥𝐵)
43anbi2i 457 . . 3 ((𝑥𝐴𝑥 ∈ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 elin 3320 . . 3 (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ (𝑥𝐴𝑥 ∈ (V ∖ 𝐵)))
6 eldif 3140 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
74, 5, 63bitr4i 212 . 2 (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
87eqriv 2174 1 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104   = wceq 1353  wcel 2148  Vcvv 2739  cdif 3128  cin 3130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-in 3137
This theorem is referenced by:  indif2  3381  difundir  3390  difindir  3392  difdif2ss  3394  difun1  3397  difdifdirss  3509  nn0supp  9230
  Copyright terms: Public domain W3C validator