Theorem ssequn1 3434

Description: A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
ssequn1	|- (A C_ B <-> (A u. B) = B)

Proof of Theorem ssequn1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bicom 191	. . . 4 |- ((x e. B <-> (x e. A \/ x e. B)) <-> ((x e. A \/ x e. B) <-> x e. B))
2		pm4.72 846	. . . 4 |- ((x e. A -> x e. B) <-> (x e. B <-> (x e. A \/ x e. B)))
3		elun 3221	. . . . 5 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
4	3	bibi1i 305	. . . 4 |- ((x e. (A u. B) <-> x e. B) <-> ((x e. A \/ x e. B) <-> x e. B))
5	1, 2, 4	3bitr4i 268	. . 3 |- ((x e. A -> x e. B) <-> (x e. (A u. B) <-> x e. B))
6	5	albii 1566	. 2 |- (A.x(x e. A -> x e. B) <-> A.x(x e. (A u. B) <-> x e. B))
7		dfss2 3263	. 2 |- (A C_ B <-> A.x(x e. A -> x e. B))
8		dfcleq 2347	. 2 |- ((A u. B) = B <-> A.x(x e. (A u. B) <-> x e. B))
9	6, 7, 8	3bitr4i 268	1 |- (A C_ B <-> (A u. B) = B)

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357  A.wal 1540   = wceq 1642   e. wcel 1710   u. cun 3208   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260

This theorem is referenced by:  ssequn2  3437  undif  3631  unsneqsn  3888  dflec2  6211