Theorem unss2 2191

Description: Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18.

Assertion

Ref	Expression
unss2	|- (A (_ B -> (C u. A) (_ (C u. B))

Proof of Theorem unss2

Step	Hyp	Ref	Expression
1		unss1 2189	. 2 |- (A (_ B -> (A u. C) (_ (B u. C))
2		uncom 2166	. 2 |- (C u. A) = (A u. C)
3		uncom 2166	. 2 |- (C u. B) = (B u. C)
4	1, 2, 3	3sstr4g 2092	1 |- (A (_ B -> (C u. A) (_ (C u. B))

Syntax hints:   -> wi 3   u. cun 2035   (_ wss 2037

This theorem is referenced by:  unss12 2192

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-in 2041  df-ss 2043

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