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Theorem dfss1 3250
Description: A frequently-used variant of subclass definition df-ss 3054. (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 3054 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 incom 3238 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
32eqeq1i 2125 . 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 1316    i^i cin 3040    C_ wss 3041
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054
This theorem is referenced by:  dfss5  3251  sseqin2  3265  onintexmid  4457  xpimasn  4957  fndmdif  5493  infiexmid  6739  ssfidc  6791  isumss  11115  znnen  11822
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