Theorem uneqri 3278

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 3277	. . 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 2174	1 |- ( A u. B ) = C

Syntax hints:   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   u. cun 3128

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134

This theorem is referenced by:  unidm  3279  uncom  3280  unass  3293  undi  3384  unab  3403  un0  3457