Theorem unssi 3178

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 267	. 2 |- ( A C_ C /\ B C_ C )
4		unss 3177	. 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 3000   C_ wss 3002

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015

This theorem is referenced by:  undifabs  3365  inundifss  3366  dmrnssfld  4711  djuun  6816  caserel  6834  ltrelxr  7610  nn0ssre  8740  nn0ssz  8831  strleun  11646