Theorem ssequn1 3193

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		pm4.72 778	. . . 4 |- ( ( x e. A -> x e. B ) <-> ( x e. B <-> ( x e. A \/ x e. B ) ) )
3		elun 3164	. . . . 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 1414	. 2 |- ( A. x ( x e. A -> x e. B ) <-> A. x ( x e. ( A u. B ) <-> x e. B ) )
7		dfss2 3036	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
8		dfcleq 2094	. 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 670   A.wal 1297   = wceq 1299   e. wcel 1448   u. cun 3019   C_ wss 3021

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-in 3027  df-ss 3034

This theorem is referenced by:  ssequn2  3196  uniop  4115  pwssunim  4144  unisuc  4273  unisucg  4274  rdgisucinc  6212  oasuc  6290  omsuc  6298  undifdc  6741