Theorem ssequn1 3343

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 140	. . . 4 |- ( ( x e. B <-> ( x e. A \/ x e. B ) ) <-> ( ( x e. A \/ x e. B ) <-> x e. B ) )
2		pm4.72 829	. . . 4 |- ( ( x e. A -> x e. B ) <-> ( x e. B <-> ( x e. A \/ x e. B ) ) )
3		elun 3314	. . . . 5 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
4	3	bibi1i 228	. . . 4 |- ( ( x e. ( A u. B ) <-> x e. B ) <-> ( ( x e. A \/ x e. B ) <-> x e. B ) )
5	1, 2, 4	3bitr4i 212	. . 3 |- ( ( x e. A -> x e. B ) <-> ( x e. ( A u. B ) <-> x e. B ) )
6	5	albii 1493	. 2 |- ( A. x ( x e. A -> x e. B ) <-> A. x ( x e. ( A u. B ) <-> x e. B ) )
7		ssalel 3181	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
8		dfcleq 2199	. 2 |- ( ( A u. B ) = B <-> A. x ( x e. ( A u. B ) <-> x e. B ) )
9	6, 7, 8	3bitr4i 212	1 |- ( A C_ B <-> ( A u. B ) = B )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 710   A.wal 1371   = wceq 1373   e. wcel 2176   u. cun 3164   C_ wss 3166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179

This theorem is referenced by:  ssequn2  3346  uniop  4300  pwssunim  4331  unisuc  4460  unisucg  4461  rdgisucinc  6471  oasuc  6550  omsuc  6558  undifdc  7021