Theorem uneqri 3346

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 3345	. . 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 184	. 2 |- ( x e. ( A u. B ) <-> x e. C )
4	3	eqriv 2226	1 |- ( A u. B ) = C

Syntax hints:   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   u. cun 3195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201

This theorem is referenced by:  unidm  3347  uncom  3348  unass  3361  undi  3452  unab  3471  un0  3525