Theorem int0el 4934

Description: The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)

Assertion

Ref	Expression
int0el	⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅)

Proof of Theorem int0el

Step	Hyp	Ref	Expression
1		intss1 4918	. 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 ⊆ ∅)
2		0ss 4352	. . 3 ⊢ ∅ ⊆ ∩ 𝐴
3	2	a1i 11	. 2 ⊢ (∅ ∈ 𝐴 → ∅ ⊆ ∩ 𝐴)
4	1, 3	eqssd 3951	1 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ⊆ wss 3901  ∅c0 4285  ∩ cint 4902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-dif 3904  df-ss 3918  df-nul 4286  df-int 4903

This theorem is referenced by:  intv  5309  inton  6376  onint0  7736  oev2  8450  cuteq0  27811  ipolub00  49234