Theorem inv1 3430

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 3327	. 2 |- ( A i^i _V ) C_ A
2		ssid 3148	. . 3 |- A C_ A
3		ssv 3150	. . 3 |- A C_ _V
4	2, 3	ssini 3330	. 2 |- A C_ ( A i^i _V )
5	1, 4	eqssi 3144	1 |- ( A i^i _V ) = A

Syntax hints:   = wceq 1335   _Vcvv 2712   i^i cin 3101

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115

This theorem is referenced by:  rint0  3846  riin0  3920  xpssres  4900  resdmdfsn  4908  imainrect  5030  xpima2m  5032  dmresv  5043