Theorem ssequn2 3337

Description: A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)

Assertion

Ref	Expression
ssequn2	|- ( A C_ B <-> ( B u. A ) = B )

Proof of Theorem ssequn2

Step	Hyp	Ref	Expression
1		ssequn1 3334	. 2 |- ( A C_ B <-> ( A u. B ) = B )
2		uncom 3308	. . 3 |- ( A u. B ) = ( B u. A )
3	2	eqeq1i 2204	. 2 |- ( ( A u. B ) = B <-> ( B u. A ) = B )
4	1, 3	bitri 184	1 |- ( A C_ B <-> ( B u. A ) = B )

Syntax hints:   <-> wb 105   = wceq 1364   u. cun 3155   C_ wss 3157

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170

This theorem is referenced by:  unabs  3395  pwssunim  4320  pwundifss  4321  oneluni  4467  relresfld  5200  relcoi1  5202  fsnunf  5765  unsnfidcel  6991  tpfidceq  7000  fidcenumlemr  7030  exmidfodomrlemim  7280  ennnfonelemhf1o  12655  lspun0  14057  plyrecj  15083  dvply2g  15086