Theorem unss1 2210

Description: Subclass law for union of classes.

Assertion

Ref	Expression
unss1	|- (A (_ B -> (A u. C) (_ (B u. C))

Proof of Theorem unss1

Step	Hyp	Ref	Expression
1		pm2.38 572	. . . 4 |- ((x e. A -> x e. B) -> ((x e. A \/ x e. C) -> (x e. B \/ x e. C)))
2		elun 2184	. . . 4 |- (x e. (A u. C) <-> (x e. A \/ x e. C))
3		elun 2184	. . . 4 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
4	1, 2, 3	3imtr4g 556	. . 3 |- ((x e. A -> x e. B) -> (x e. (A u. C) -> x e. (B u. C)))
5	4	19.20i 996	. 2 |- (A.x(x e. A -> x e. B) -> A.x(x e. (A u. C) -> x e. (B u. C)))
6		dfss2 2069	. 2 |- (A (_ B <-> A.x(x e. A -> x e. B))
7		dfss2 2069	. 2 |- ((A u. C) (_ (B u. C) <-> A.x(x e. (A u. C) -> x e. (B u. C)))
8	5, 6, 7	3imtr4i 219	1 |- (A (_ B -> (A u. C) (_ (B u. C))

Syntax hints:   -> wi 3   \/ wo 222  A.wal 958   e. wcel 962   u. cun 2056   (_ wss 2058

This theorem is referenced by:  unss2 2212  unss12 2213  eldifpw 2926

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-v 1819  df-un 2061  df-in 2062  df-ss 2064

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