Theorem difexg 4103

Description: The difference of two sets is a set. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
difexg	⊢ ((A ∈ V ∧ B ∈ W) → (A ∖ B) ∈ V)

Proof of Theorem difexg

Step	Hyp	Ref	Expression
1		df-dif 3216	. 2 ⊢ (A ∖ B) = (A ∩ ∼ B)
2		complexg 4100	. . 3 ⊢ (B ∈ W → ∼ B ∈ V)
3		inexg 4101	. . 3 ⊢ ((A ∈ V ∧ ∼ B ∈ V) → (A ∩ ∼ B) ∈ V)
4	2, 3	sylan2 460	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → (A ∩ ∼ B) ∈ V)
5	1, 4	syl5eqel 2437	1 ⊢ ((A ∈ V ∧ B ∈ W) → (A ∖ B) ∈ V)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∩ cin 3209

This theorem was proved from 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  ax-nin 4079

This theorem 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-dif 3216

This theorem is referenced by:  symdifexg  4104  difex  4108  imagekexg  4312  pwexg  4329  fullfunexg  5860  fnfreclem1  6318