Theorem inundif 3629

Description: The intersection and class difference of a class with another class unite to give the original class. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
inundif	⊢ ((A ∩ B) ∪ (A ∖ B)) = A

Proof of Theorem inundif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3220	. . . 4 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
2		eldif 3222	. . . 4 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
3	1, 2	orbi12i 507	. . 3 ⊢ ((x ∈ (A ∩ B) ∨ x ∈ (A ∖ B)) ↔ ((x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ ¬ x ∈ B)))
4		pm4.42 926	. . 3 ⊢ (x ∈ A ↔ ((x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ ¬ x ∈ B)))
5	3, 4	bitr4i 243	. 2 ⊢ ((x ∈ (A ∩ B) ∨ x ∈ (A ∖ B)) ↔ x ∈ A)
6	5	uneqri 3407	1 ⊢ ((A ∩ B) ∪ (A ∖ B)) = A

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216

This theorem is used by:  phialllem2  4618  sbthlem1  6204