Theorem difdif 1595

Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22.

Assertion

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

Proof of Theorem difdif

Step	Hyp	Ref	Expression
1		eldif 1496	. . . . . 6 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A))
2	1	negbii 162	. . . . 5 ⊢ (¬ x ∈ (B ∖ A) ↔ ¬ (x ∈ B ∧ ¬ x ∈ A))
3		iman 205	. . . . 5 ⊢ ((x ∈ B → x ∈ A) ↔ ¬ (x ∈ B ∧ ¬ x ∈ A))
4	2, 3	bitr4 154	. . . 4 ⊢ (¬ x ∈ (B ∖ A) ↔ (x ∈ B → x ∈ A))
5	4	anbi2i 367	. . 3 ⊢ ((x ∈ A ∧ ¬ x ∈ (B ∖ A)) ↔ (x ∈ A ∧ (x ∈ B → x ∈ A)))
6		pm4.45im 267	. . 3 ⊢ (x ∈ A ↔ (x ∈ A ∧ (x ∈ B → x ∈ A)))
7	5, 6	bitr4 154	. 2 ⊢ ((x ∈ A ∧ ¬ x ∈ (B ∖ A)) ↔ x ∈ A)
8	7	difeqri 1589	1 ⊢ (A ∖ (B ∖ A)) = A

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∖ cdif 1484

This theorem is referenced by:  dif0 1756

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489

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