Theorem inv1 3812

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 3718	. 2 |- ( A i^i _V ) C_ A
2		ssid 3523	. . 3 |- A C_ A
3		ssv 3524	. . 3 |- A C_ _V
4	2, 3	ssini 3721	. 2 |- A C_ ( A i^i _V )
5	1, 4	eqssi 3520	1 |- ( A i^i _V ) = A

Syntax hints:   = wceq 1379   _Vcvv 3113   i^i cin 3475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-in 3483  df-ss 3490

This theorem is referenced by:  undif1  3902  dfif4  3954  rint0  4322  iinrab2  4388  riin0  4399  xpriindi  5139  xpssres  5308  resdmdfsn  5319  imainrect  5448  xpima  5449  dmresv  5465  curry1  6875  curry2  6878  fpar  6887  oev2  7173  hashresfn  12381  dmhashres  12382  gsumxp  16807  gsumxpOLD  16809  pjpm  18534  ptbasfi  19845  mbfmcst  27898  0rrv  28058  pol0N  34723