Theorem int0el 5003

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 4987	. 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 ⊆ ∅)
2		0ss 4423	. . 3 ⊢ ∅ ⊆ ∩ 𝐴
3	2	a1i 11	. 2 ⊢ (∅ ∈ 𝐴 → ∅ ⊆ ∩ 𝐴)
4	1, 3	eqssd 4026	1 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  ∅c0 4352  ∩ cint 4970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-ss 3993  df-nul 4353  df-int 4971

This theorem is referenced by:  intv  5382  inton  6453  onint0  7827  oev2  8579  cuteq0  27895  ipolub00  48665