Theorem difdif 3063

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		simpl 102	. . 3 ⊢ ((x ∈ A ∧ ¬ x ∈ (B ∖ A)) → x ∈ A)
2		pm4.45im 317	. . . 4 ⊢ (x ∈ A ↔ (x ∈ A ∧ (x ∈ B → x ∈ A)))
3		imanim 784	. . . . . 6 ⊢ ((x ∈ B → x ∈ A) → ¬ (x ∈ B ∧ ¬ x ∈ A))
4		eldif 2921	. . . . . 6 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A))
5	3, 4	sylnibr 601	. . . . 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 3058	1 ⊢ (A ∖ (B ∖ A)) = A

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   ∖ cdif 2908

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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914

This theorem is referenced by:  dif0  3288