Theorem in0 3577

Description: The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
in0	⊢ (A ∩ ∅) = ∅

Proof of Theorem in0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3555	. . . 4 ⊢ ¬ x ∈ ∅
2	1	bianfi 891	. . 3 ⊢ (x ∈ ∅ ↔ (x ∈ A ∧ x ∈ ∅))
3	2	bicomi 193	. 2 ⊢ ((x ∈ A ∧ x ∈ ∅) ↔ x ∈ ∅)
4	3	ineqri 3450	1 ⊢ (A ∩ ∅) = ∅

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∩ cin 3209  ∅c0 3551

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-nul 3552

This theorem is referenced by:  inindif  4076  addcid1  4406  res0  4978