Theorem ssequn1 3292

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 139	. . . 4 |- ( ( x e. B <-> ( x e. A \/ x e. B ) ) <-> ( ( x e. A \/ x e. B ) <-> x e. B ) )
2		pm4.72 817	. . . 4 |- ( ( x e. A -> x e. B ) <-> ( x e. B <-> ( x e. A \/ x e. B ) ) )
3		elun 3263	. . . . 5 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
4	3	bibi1i 227	. . . 4 |- ( ( x e. ( A u. B ) <-> x e. B ) <-> ( ( x e. A \/ x e. B ) <-> x e. B ) )
5	1, 2, 4	3bitr4i 211	. . 3 |- ( ( x e. A -> x e. B ) <-> ( x e. ( A u. B ) <-> x e. B ) )
6	5	albii 1458	. 2 |- ( A. x ( x e. A -> x e. B ) <-> A. x ( x e. ( A u. B ) <-> x e. B ) )
7		dfss2 3131	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
8		dfcleq 2159	. 2 |- ( ( A u. B ) = B <-> A. x ( x e. ( A u. B ) <-> x e. B ) )
9	6, 7, 8	3bitr4i 211	1 |- ( A C_ B <-> ( A u. B ) = B )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 698   A.wal 1341   = wceq 1343   e. wcel 2136   u. cun 3114   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129

This theorem is referenced by:  ssequn2  3295  uniop  4233  pwssunim  4262  unisuc  4391  unisucg  4392  rdgisucinc  6353  oasuc  6432  omsuc  6440  undifdc  6889