Theorem unssi 2781

Description: An inference that 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 307	. 2 |- (A C_ C /\ B C_ C)
4		unss 2780	. 2 |- ((A C_ C /\ B C_ C) <-> (A u. B) C_ C)
5	3, 4	mpbi 206	1 |- (A u. B) C_ C

Syntax hints:   /\ wa 240   u. cun 2591   C_ wss 2593

This theorem is referenced by:  dmrnssfld 4205  fparlem3 5083  fparlem4 5084  ider 5326  rankun 5802  nn0ssre 7312  nn0ssz 7361  cdrci 10182  shsleji 10971  shlubi 10979  shsumval3i 10994  shjshsi 11048  spanuni 11100  sshhococi 11102  osumi 11221

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-in 2603  df-ss 2605

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