Theorem ssequn1 3251

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		pm4.72 813	. . . 4 |- ( ( x e. A -> x e. B ) <-> ( x e. B <-> ( x e. A \/ x e. B ) ) )
3		elun 3222	. . . . 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 1447	. 2 |- ( A. x ( x e. A -> x e. B ) <-> A. x ( x e. ( A u. B ) <-> x e. B ) )
7		dfss2 3091	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
8		dfcleq 2134	. 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 1330   = wceq 1332   e. wcel 1481   u. cun 3074   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089

This theorem is referenced by:  ssequn2  3254  uniop  4185  pwssunim  4214  unisuc  4343  unisucg  4344  rdgisucinc  6290  oasuc  6368  omsuc  6376  undifdc  6820