Theorem unssi 1633

Description: An inference that the union of two subclasses is a subclass. Contributed by Raph Levien, 10-Dec-02.

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 234	. 2 |- (A (_ C /\ B (_ C)
4		unss 1632	. 2 |- ((A (_ C /\ B (_ C) <-> (A u. B) (_ C)
5	3, 4	mpbi 164	1 |- (A u. B) (_ C

Syntax hints:   /\ wa 196   u. cun 1485   (_ wss 1487

This theorem is referenced by:  rankun 3535  nn0ssre 4538  nn0ssz 4578  shslej 5339  shlub 5347  shsumval3 5362  shjshs 5412  spanun 5450  sshhococ 5451  osum 5538

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492

metamath.org