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Theorem dfss1 3202
Description: A frequently-used variant of subclass definition df-ss 3010. (Contributed by NM, 10-Jan-2015.)
Assertion
Ref Expression
dfss1  |-  ( A 
C_  B  <->  ( B  i^i  A )  =  A )

Proof of Theorem dfss1
StepHypRef Expression
1 df-ss 3010 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 incom 3190 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
32eqeq1i 2095 . 2  |-  ( ( A  i^i  B )  =  A  <->  ( B  i^i  A )  =  A )
41, 3bitri 182 1  |-  ( A 
C_  B  <->  ( B  i^i  A )  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1289    i^i cin 2996    C_ wss 2997
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010
This theorem is referenced by:  dfss5  3203  sseqin2  3217  onintexmid  4378  xpimasn  4866  fndmdif  5388  infiexmid  6573  ssfidc  6623  isumss  10747  znnen  11293
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