Theorem int0el 4946

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 4930	. 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 ⊆ ∅)
2		0ss 4357	. . 3 ⊢ ∅ ⊆ ∩ 𝐴
3	2	a1i 11	. 2 ⊢ (∅ ∈ 𝐴 → ∅ ⊆ ∩ 𝐴)
4	1, 3	eqssd 3955	1 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906  ∅c0 4286  ∩ cint 4914

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3909  df-ss 3923  df-nul 4287  df-int 4915

This theorem is used by:  intv  5337  inton  6424  onint0  7792  oev2  8510  cuteq0  28037  nmulr0  36700  ipolub00  49804