Theorem in0 1722

Description: The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26.

Assertion

Ref	Expression
in0	⊢ (A ∩ ∅) = ∅

Proof of Theorem in0

Step	Hyp	Ref	Expression
1		noel 1711	. . . 4 ⊢ ¬ x ∈ ∅
2	1	bianfi 553	. . 3 ⊢ (x ∈ ∅ ↔ (x ∈ A ∧ x ∈ ∅))
3	2	bicomi 150	. 2 ⊢ ((x ∈ A ∧ x ∈ ∅) ↔ x ∈ ∅)
4	3	ineqri 1637	1 ⊢ (A ∩ ∅) = ∅

Syntax hints:   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∩ cin 1486  ∅c0 1707

This theorem is referenced by:  0ex 1745  difin0 1759  res0 2578  resdisj 2656

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-nul 1708

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