Theorem unssi 3246

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 3245	. 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 3064   C_ wss 3066

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079

This theorem is referenced by:  undifabs  3434  inundifss  3435  dmrnssfld  4797  caserel  6965  ltrelxr  7818  nn0ssre  8974  nn0ssz  9065  strleun  12037