Theorem uni0c 2524

Description: The union of a set is empty iff all of its members are empty.

Assertion

Ref	Expression
uni0c	|- (U.A = (/) <-> A.x e. A x = (/))

Distinct variable group:   x,A

Proof of Theorem uni0c

Step	Hyp	Ref	Expression
1		uni0b 2523	. 2 |- (U.A = (/) <-> A (_ {(/)})
2		dfss3 2059	. 2 |- (A (_ {(/)} <-> A.x e. A x e. {(/)})
3		elsn 2421	. . 3 |- (x e. {(/)} <-> x = (/))
4	3	ralbii 1667	. 2 |- (A.x e. A x e. {(/)} <-> A.x e. A x = (/))
5	1, 2, 4	3bitr 177	1 |- (U.A = (/) <-> A.x e. A x = (/))

Syntax hints:   <-> wb 146   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047  (/)c0 2280  {csn 2409  U.cuni 2503

This theorem is referenced by:  fctopOLD 7650

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-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-uni 2504

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