Theorem int0el 4938

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 4922	. 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 ⊆ ∅)
2		0ss 4349	. . 3 ⊢ ∅ ⊆ ∩ 𝐴
3	2	a1i 11	. 2 ⊢ (∅ ∈ 𝐴 → ∅ ⊆ ∩ 𝐴)
4	1, 3	eqssd 3925	1 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895  ∅c0 4278  ∩ cint 4906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-dif 4047  df-nul 4279  df-int 4907

This theorem is used by:  intv  5325  inton  6422  onint0  7806  oev2  8535  cuteq0  28207  nmulr0  37037  ipolub00  50201