Theorem intmin3 3851

Description: Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)

Hypotheses

Ref	Expression
intmin3.2	|- ( x = A -> ( ph <-> ps ) )
intmin3.3	|- ps

Assertion

Ref	Expression
intmin3	|- ( A e. V -> |^| { x | ph } C_ A )

Distinct variable groups:   x, A   ps, x

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem intmin3

Step	Hyp	Ref	Expression
1		intmin3.3	. . 3 |- ps
2		intmin3.2	. . . 4 |- ( x = A -> ( ph <-> ps ) )
3	2	elabg 2872	. . 3 |- ( A e. V -> ( A e. { x | ph } <-> ps ) )
4	1, 3	mpbiri 167	. 2 |- ( A e. V -> A e. { x | ph } )
5		intss1 3839	. 2 |- ( A e. { x | ph } -> |^| { x | ph } C_ A )
6	4, 5	syl 14	1 |- ( A e. V -> |^| { x | ph } C_ A )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   {cab 2151   C_ wss 3116   |^|cint 3824

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-int 3825

This theorem is referenced by:  intid  4202