Theorem in0 2298

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 2284	. . . 4 |- -. x e. (/)
2	1	bianfi 737	. . 3 |- (x e. (/) <-> (x e. A /\ x e. (/)))
3	2	bicomi 172	. 2 |- ((x e. A /\ x e. (/)) <-> x e. (/))
4	3	ineqri 2209	1 |- (A i^i (/)) = (/)

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958   i^i cin 2046  (/)c0 2280

This theorem is referenced by:  difin0 2338  res0 3371  resdisj 3471  oev2 4162  sn0top 7647  indistop 7648  fctopOLD 7650  cctop 7652  neiopne 10474  rcfpfillem5 10593  rcfpfillem5OLD 10594  emhgrat 10775

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  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 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-dif 2049  df-in 2051  df-nul 2281

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