Theorem difdif 3046

Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)

Assertion

Ref	Expression
difdif	⊢ (A ∖ (B ∖ A)) = A

Proof of Theorem difdif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-ia1 99	. . 3 ⊢ ((x ∈ A ∧ ¬ x ∈ (B ∖ A)) → x ∈ A)
2		pm4.45im 317	. . . 4 ⊢ (x ∈ A ↔ (x ∈ A ∧ (x ∈ B → x ∈ A)))
3		imanim 778	. . . . . 6 ⊢ ((x ∈ B → x ∈ A) → ¬ (x ∈ B ∧ ¬ x ∈ A))
4		eldif 2904	. . . . . 6 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A))
5	3, 4	sylnibr 589	. . . . 5 ⊢ ((x ∈ B → x ∈ A) → ¬ x ∈ (B ∖ A))
6	5	anim2i 324	. . . 4 ⊢ ((x ∈ A ∧ (x ∈ B → x ∈ A)) → (x ∈ A ∧ ¬ x ∈ (B ∖ A)))
7	2, 6	sylbi 114	. . 3 ⊢ (x ∈ A → (x ∈ A ∧ ¬ x ∈ (B ∖ A)))
8	1, 7	impbii 117	. 2 ⊢ ((x ∈ A ∧ ¬ x ∈ (B ∖ A)) ↔ x ∈ A)
9	8	difeqri 3041	1 ⊢ (A ∖ (B ∖ A)) = A

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374   ∖ cdif 2891

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897

This theorem is referenced by:  dif0  3271