Theorem invdif 3416

Description: Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
invdif	⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Proof of Theorem invdif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2776	. . . . 5 ⊢ 𝑥 ∈ V
2		eldif 3176	. . . . 5 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐵))
3	1, 2	mpbiran 943	. . . 4 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥 ∈ 𝐵)
4	3	anbi2i 457	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (V ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5		elin 3357	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (V ∖ 𝐵)))
6		eldif 3176	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
7	4, 5, 6	3bitr4i 212	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ (V ∖ 𝐵)) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))
8	7	eqriv 2203	1 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1373   ∈ wcel 2177  Vcvv 2773   ∖ cdif 3164   ∩ cin 3166

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3169  df-in 3173

This theorem is referenced by:  indif2  3418  difundir  3427  difindir  3429  difdif2ss  3431  difun1  3434  difdifdirss  3546  nn0supp  9354