Theorem elnin 3225

Description: Membership in anti-intersection. (Contributed by SF, 10-Jan-2015.)

Hypothesis

Ref	Expression
elbool.1	⊢ A ∈ V

Assertion

Ref	Expression
elnin	⊢ (A ∈ (B ⩃ C) ↔ (A ∈ B ⊼ A ∈ C))

Proof of Theorem elnin

Step	Hyp	Ref	Expression
1		elbool.1	. 2 ⊢ A ∈ V
2		elning 3218	. 2 ⊢ (A ∈ V → (A ∈ (B ⩃ C) ↔ (A ∈ B ⊼ A ∈ C)))
3	1, 2	ax-mp 5	1 ⊢ (A ∈ (B ⩃ C) ↔ (A ∈ B ⊼ A ∈ C))

Syntax hints:   ↔ wb 176   ⊼ wnan 1287   ∈ wcel 1710  Vcvv 2860   ⩃ cnin 3205

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

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

This theorem is referenced by:  nincom  3227  nincompl  4073  ninexg  4098