Theorem inv1 3549

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 3445	. 2 |- ( A i^i _V ) C_ A
2		ssid 3262	. . 3 |- A C_ A
3		ssv 3264	. . 3 |- A C_ _V
4	2, 3	ssini 3448	. 2 |- A C_ ( A i^i _V )
5	1, 4	eqssi 3258	1 |- ( A i^i _V ) = A

Syntax hints:   = wceq 1398   _Vcvv 2815   i^i cin 3213

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-ss 3227

This theorem is referenced by:  rint0  3993  riin0  4068  xpssres  5078  resdmdfsn  5086  imainrect  5213  xpima2m  5215  dmresv  5226