Theorem inv1 3460

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 3356	. 2 |- ( A i^i _V ) C_ A
2		ssid 3176	. . 3 |- A C_ A
3		ssv 3178	. . 3 |- A C_ _V
4	2, 3	ssini 3359	. 2 |- A C_ ( A i^i _V )
5	1, 4	eqssi 3172	1 |- ( A i^i _V ) = A

Syntax hints:   = wceq 1353   _Vcvv 2738   i^i cin 3129

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136  df-ss 3143

This theorem is referenced by:  rint0  3884  riin0  3959  xpssres  4943  resdmdfsn  4951  imainrect  5075  xpima2m  5077  dmresv  5088