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Theorem dfss1 3275
Description: A frequently-used variant of subclass definition df-ss 3079. (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 3079 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 incom 3263 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
32eqeq1i 2145 . 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 1331    i^i cin 3065    C_ wss 3066
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079
This theorem is referenced by:  dfss5  3276  sseqin2  3290  onintexmid  4482  xpimasn  4982  fndmdif  5518  infiexmid  6764  ssfidc  6816  isumss  11153  znnen  11900
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