Theorem unssi 2205

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
unssi.2	|- B (_ C

Assertion

Ref	Expression
unssi	|- (A u. B) (_ C

Proof of Theorem unssi

Step	Hyp	Ref	Expression
1		unssi.1	. . 3 |- A (_ C
2		unssi.2	. . 3 |- B (_ C
3	1, 2	pm3.2i 285	. 2 |- (A (_ C /\ B (_ C)
4		unss 2204	. 2 |- ((A (_ C /\ B (_ C) <-> (A u. B) (_ C)
5	3, 4	mpbi 189	1 |- (A u. B) (_ C

Syntax hints:   /\ wa 223   u. cun 2045   (_ wss 2047

This theorem is referenced by:  dmrnssfld 3357  rankun 4691  nn0ssre 6103  nn0ssz 6152  shslej 9338  shlub 9346  shsumval3 9361  shjshs 9415  spanun 9467  sshhococ 9469  osum 9586  cdrci 10494

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-in 2051  df-ss 2053

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