Theorem unss1 2170

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		pm2.38 567	. . . 4 |- ((x e. A -> x e. B) -> ((x e. A \/ x e. C) -> (x e. B \/ x e. C)))
2		elun 2144	. . . 4 |- (x e. (A u. C) <-> (x e. A \/ x e. C))
3		elun 2144	. . . 4 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
4	1, 2, 3	3imtr4g 551	. . 3 |- ((x e. A -> x e. B) -> (x e. (A u. C) -> x e. (B u. C)))
5	4	19.20i 968	. 2 |- (A.x(x e. A -> x e. B) -> A.x(x e. (A u. C) -> x e. (B u. C)))
6		dfss2 2029	. 2 |- (A (_ B <-> A.x(x e. A -> x e. B))
7		dfss2 2029	. 2 |- ((A u. C) (_ (B u. C) <-> A.x(x e. (A u. C) -> x e. (B u. C)))
8	5, 6, 7	3imtr4 219	1 |- (A (_ B -> (A u. C) (_ (B u. C))

Syntax hints:   -> wi 3   \/ wo 222  A.wal 950   e. wcel 1105   u. cun 2016   (_ wss 2018

This theorem is referenced by:  unss2 2172  unss12 2173  eldifpw 2873

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-un 2021  df-in 2022  df-ss 2024

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