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 i^i (/)) = (/)

Proof of Theorem in0

Step	Hyp	Ref	Expression
1		noel 1711	. . . 4 |- -. x e. (/)
2	1	bianfi 553	. . 3 |- (x e. (/) <-> (x e. A /\ x e. (/)))
3	2	bicomi 150	. 2 |- ((x e. A /\ x e. (/)) <-> x e. (/))
4	3	ineqri 1637	1 |- (A i^i (/)) = (/)

Syntax hints:   /\ wa 196   = wceq 1091   e. wcel 1092   i^i 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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