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

Step	Hyp	Ref	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
4	2, 3	ssini 3265	. 2 |- A C_ ( A i^i _V )
5	1, 4	eqssi 3079	1 |- ( A i^i _V ) = A

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