Theorem int0el 4936

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 4920	. 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 ⊆ ∅)
2		0ss 4354	. . 3 ⊢ ∅ ⊆ ∩ 𝐴
3	2	a1i 11	. 2 ⊢ (∅ ∈ 𝐴 → ∅ ⊆ ∩ 𝐴)
4	1, 3	eqssd 3953	1 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  ∅c0 4287  ∩ cint 4904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-ss 3920  df-nul 4288  df-int 4905

This theorem is referenced by:  intv  5311  inton  6384  onint0  7746  oev2  8460  cuteq0  27823  ipolub00  49346