Theorem ssequn1OLD 2776

Description: A relationship between subclass and union. Theorem 26 of [Suppes] p. 27.

Assertion

Ref	Expression
ssequn1OLD	|- (A C_ B <-> (A u. B) = B)

Proof of Theorem ssequn1OLD

Step	Hyp	Ref	Expression
1		df-un 2600	. . 3 |- (A u. B) = {x | (x e. A \/ x e. B)}
2	1	eqeq2i 1894	. 2 |- (B = (A u. B) <-> B = {x | (x e. A \/ x e. B)})
3		eqcom 1886	. 2 |- ((A u. B) = B <-> B = (A u. B))
4		pm4.72 703	. . . 4 |- ((x e. A -> x e. B) <-> (x e. B <-> (x e. A \/ x e. B)))
5	4	albii 1346	. . 3 |- (A.x(x e. A -> x e. B) <-> A.x(x e. B <-> (x e. A \/ x e. B)))
6		dfss2 2610	. . 3 |- (A C_ B <-> A.x(x e. A -> x e. B))
7		abeq2 1999	. . 3 |- (B = {x | (x e. A \/ x e. B)} <-> A.x(x e. B <-> (x e. A \/ x e. B)))
8	5, 6, 7	3bitr4i 200	. 2 |- (A C_ B <-> B = {x | (x e. A \/ x e. B)})
9	2, 3, 8	3bitr4ri 201	1 |- (A C_ B <-> (A u. B) = B)

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871   u. cun 2591   C_ wss 2593

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-un 2600  df-in 2603  df-ss 2605

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