Theorem ssequn1 3298

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		pm4.72 823	. . . 4 |- ( ( x e. A -> x e. B ) <-> ( x e. B <-> ( x e. A \/ x e. B ) ) )
3		elun 3269	. . . . 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 1464	. 2 |- ( A. x ( x e. A -> x e. B ) <-> A. x ( x e. ( A u. B ) <-> x e. B ) )
7		dfss2 3137	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
8		dfcleq 2165	. 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 704   A.wal 1347   = wceq 1349   e. wcel 2142   u. cun 3120   C_ wss 3122

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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-v 2733  df-un 3126  df-in 3128  df-ss 3135

This theorem is referenced by:  ssequn2  3301  uniop  4241  pwssunim  4270  unisuc  4399  unisucg  4400  rdgisucinc  6368  oasuc  6447  omsuc  6455  undifdc  6905