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Theorem dfss1 3311
Description: A frequently-used variant of subclass definition df-ss 3115. (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 3115 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 incom 3299 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
32eqeq1i 2165 . 2  |-  ( ( A  i^i  B )  =  A  <->  ( B  i^i  A )  =  A )
41, 3bitri 183 1  |-  ( A 
C_  B  <->  ( B  i^i  A )  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1335    i^i cin 3101    C_ wss 3102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115
This theorem is referenced by:  dfss5  3312  sseqin2  3326  onintexmid  4533  xpimasn  5035  fndmdif  5573  infiexmid  6823  ssfidc  6880  isumss  11292  znnen  12169
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