Theorem dfin4 1673

Description: Alternate definition of the union of two classes. Exercise 4.10(q) of [Mendelson] p. 231.

Assertion

Ref	Expression
dfin4	⊢ (A ∩ B) = (A ∖ (A ∖ B))

Proof of Theorem dfin4

Step	Hyp	Ref	Expression
1		inss1 1657	. . 3 ⊢ (A ∩ B) ⊆ A
2		dfss4 1667	. . 3 ⊢ ((A ∩ B) ⊆ A ↔ (A ∖ (A ∖ (A ∩ B))) = (A ∩ B))
3	1, 2	mpbi 164	. 2 ⊢ (A ∖ (A ∖ (A ∩ B))) = (A ∩ B)
4		difin 1670	. . 3 ⊢ (A ∖ (A ∩ B)) = (A ∖ B)
5	4	difeq2i 1585	. 2 ⊢ (A ∖ (A ∖ (A ∩ B))) = (A ∖ (A ∖ B))
6	3, 5	eqtr3 1121	1 ⊢ (A ∩ B) = (A ∖ (A ∖ B))

Syntax hints:   = wceq 1091   ∖ cdif 1484   ∩ cin 1486   ⊆ wss 1487

This theorem is referenced by:  indif 1675

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  df-in 1491  df-ss 1492

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