Theorem intmin3 3897

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 2906	. . 3 |- ( A e. V -> ( A e. { x | ph } <-> ps ) )
4	1, 3	mpbiri 168	. 2 |- ( A e. V -> A e. { x | ph } )
5		intss1 3885	. 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 105   = wceq 1364   e. wcel 2164   {cab 2179   C_ wss 3153   |^|cint 3870

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-int 3871

This theorem is referenced by:  intid  4253