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Theorem ssequn2 3308
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
StepHypRef Expression
1 ssequn1 3305 . 2  |-  ( A 
C_  B  <->  ( A  u.  B )  =  B )
2 uncom 3279 . . 3  |-  ( A  u.  B )  =  ( B  u.  A
)
32eqeq1i 2185 . 2  |-  ( ( A  u.  B )  =  B  <->  ( B  u.  A )  =  B )
41, 3bitri 184 1  |-  ( A 
C_  B  <->  ( B  u.  A )  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1353    u. cun 3127    C_ wss 3129
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142
This theorem is referenced by:  unabs  3366  pwssunim  4284  pwundifss  4285  oneluni  4431  relresfld  5158  relcoi1  5160  fsnunf  5716  unsnfidcel  6919  fidcenumlemr  6953  exmidfodomrlemim  7199  ennnfonelemhf1o  12408
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