Theorem unssi 3382

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 272	. 2 |- ( A C_ C /\ B C_ C )
4		unss 3381	. 2 |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C )
5	3, 4	mpbi 145	1 |- ( A u. B ) C_ C

Syntax hints:   /\ wa 104   u. cun 3198   C_ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213

This theorem is referenced by:  undifabs  3571  inundifss  3572  dmrnssfld  4995  caserel  7285  ltrelxr  8239  nn0ssre  9405  nn0ssz  9496  strleun  13186