Theorem unssi 3147

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 266	. 2 |- ( A C_ C /\ B C_ C )
4		unss 3146	. 2 |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C )
5	3, 4	mpbi 143	1 |- ( A u. B ) C_ C

Syntax hints:   /\ wa 102   u. cun 2971   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986

This theorem is referenced by:  undifabs  3320  inundifss  3321  dmrnssfld  4613  ltrelxr  7173  nn0ssre  8292  nn0ssz  8369