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Theorem inv1 3365
Description: The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
inv1  |-  ( A  i^i  _V )  =  A

Proof of Theorem inv1
StepHypRef Expression
1 inss1 3262 . 2  |-  ( A  i^i  _V )  C_  A
2 ssid 3083 . . 3  |-  A  C_  A
3 ssv 3085 . . 3  |-  A  C_  _V
42, 3ssini 3265 . 2  |-  A  C_  ( A  i^i  _V )
51, 4eqssi 3079 1  |-  ( A  i^i  _V )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1314   _Vcvv 2657    i^i cin 3036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-in 3043  df-ss 3050
This theorem is referenced by:  rint0  3776  riin0  3850  xpssres  4812  resdmdfsn  4820  imainrect  4942  xpima2m  4944  dmresv  4955
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