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Theorem ssequn2 3159
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 3156 . 2  |-  ( A 
C_  B  <->  ( A  u.  B )  =  B )
2 uncom 3130 . . 3  |-  ( A  u.  B )  =  ( B  u.  A
)
32eqeq1i 2092 . 2  |-  ( ( A  u.  B )  =  B  <->  ( B  u.  A )  =  B )
41, 3bitri 182 1  |-  ( A 
C_  B  <->  ( B  u.  A )  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1287    u. cun 2984    C_ wss 2986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-un 2990  df-in 2992  df-ss 2999
This theorem is referenced by:  unabs  3216  pwssunim  4078  pwundifss  4079  oneluni  4225  relresfld  4917  relcoi1  4919  fsnunf  5441  unsnfidcel  6561  exmidfodomrlemim  6748
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