Theorem unssi 3296

Description: An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)

Hypotheses

Ref	Expression
unssi.1	|- A C_ C
unssi.2	|- B C_ C

Assertion

Ref	Expression
unssi	|- ( A u. B ) C_ C

Proof of Theorem unssi

Step	Hyp	Ref	Expression
1		unssi.1	. . 3 |- A C_ C
2		unssi.2	. . 3 |- B C_ C
3	1, 2	pm3.2i 270	. 2 |- ( A C_ C /\ B C_ C )
4		unss 3295	. 2 |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C )
5	3, 4	mpbi 144	1 |- ( A u. B ) C_ C

Syntax hints:   /\ wa 103   u. cun 3113   C_ wss 3115

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-un 3119  df-in 3121  df-ss 3128

This theorem is referenced by:  undifabs  3484  inundifss  3485  dmrnssfld  4866  caserel  7048  ltrelxr  7955  nn0ssre  9114  nn0ssz  9205  strleun  12479