Theorem uneqri 3114

Description: Inference from membership to union. (Contributed by NM, 5-Aug-1993.)

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 3113	. . 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	bitri 182	. 2 |- ( x e. ( A u. B ) <-> x e. C )
4	3	eqriv 2078	1 |- ( A u. B ) = C

Syntax hints:   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   u. cun 2971

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977

This theorem is referenced by:  unidm  3115  uncom  3116  unass  3129  undi  3212  unab  3231  un0  3278