Theorem int0el 3958

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 3942	. 2 ⊢ (∅ ∈ A → ∩A ⊆ ∅)
2		0ss 3580	. . 3 ⊢ ∅ ⊆ ∩A
3	2	a1i 10	. 2 ⊢ (∅ ∈ A → ∅ ⊆ ∩A)
4	1, 3	eqssd 3290	1 ⊢ (∅ ∈ A → ∩A = ∅)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710   ⊆ wss 3258  ∅c0 3551  ∩cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-int 3928

This theorem is referenced by: (None)