Theorem intmin3 3642

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 2688	. . 3 |- ( A e. V -> ( A e. { x | ph } <-> ps ) )
4	1, 3	mpbiri 157	. 2 |- ( A e. V -> A e. { x | ph } )
5		intss1 3630	. 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 98   = wceq 1243   e. wcel 1393   {cab 2026   C_ wss 2917   |^|cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-int 3616

This theorem is referenced by:  intid  3960