Theorem in0 2296

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 2282	. . . 4 |- -. x e. (/)
2	1	bianfi 736	. . 3 |- (x e. (/) <-> (x e. A /\ x e. (/)))
3	2	bicomi 172	. 2 |- ((x e. A /\ x e. (/)) <-> x e. (/))
4	3	ineqri 2207	1 |- (A i^i (/)) = (/)

Syntax hints:   /\ wa 223   = wceq 955   e. wcel 957   i^i cin 2044  (/)c0 2278

This theorem is referenced by:  difin0 2336  res0 3368  resdisj 3468  oev2 4159  sn0top 7626  indistop 7627  fctop 7629  cctop 7631  neiopne 10462  rcfpfillem5 10550  emhgrat 10719

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1810  df-dif 2047  df-in 2049  df-nul 2279

Copyright terms: Public domain