Theorem uneqri 3269

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 3268	. . 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 183	. 2 |- ( x e. ( A u. B ) <-> x e. C )
4	3	eqriv 2167	1 |- ( A u. B ) = C

Syntax hints:   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   u. cun 3119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125

This theorem is referenced by:  unidm  3270  uncom  3271  unass  3284  undi  3375  unab  3394  un0  3448