Theorem uneqri 2174

Description: Inference from membership to union.

Hypothesis

Ref	Expression
uneqri.1	|- ((x e. A \/ x e. B) <-> x e. C)

Assertion

Ref	Expression
uneqri	|- (A u. B) = C

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem uneqri

Step	Hyp	Ref	Expression
1		elun 2173	. . 3 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
2		uneqri.1	. . 3 |- ((x e. A \/ x e. B) <-> x e. C)
3	1, 2	bitr 173	. 2 |- (x e. (A u. B) <-> x e. C)
4	3	eqriv 1474	1 |- (A u. B) = C

Syntax hints:   <-> wb 146   \/ wo 222   = wceq 956   e. wcel 958   u. cun 2045

This theorem is referenced by:  unidm 2175  unass 2187  un0 2297

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-un 2050

Copyright terms: Public domain