Theorem inv1 3505

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 3401	. 2 |- ( A i^i _V ) C_ A
2		ssid 3221	. . 3 |- A C_ A
3		ssv 3223	. . 3 |- A C_ _V
4	2, 3	ssini 3404	. 2 |- A C_ ( A i^i _V )
5	1, 4	eqssi 3217	1 |- ( A i^i _V ) = A

Syntax hints:   = wceq 1373   _Vcvv 2776   i^i cin 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187

This theorem is referenced by:  rint0  3938  riin0  4013  xpssres  5013  resdmdfsn  5021  imainrect  5147  xpima2m  5149  dmresv  5160