Theorem intminss 3849

Description: Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)

Hypothesis

Ref	Expression
intminss.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
intminss	|- ( ( A e. B /\ ps ) -> |^| { x e. B | ph } C_ A )

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem intminss

Step	Hyp	Ref	Expression
1		intminss.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
2	1	elrab 2882	. 2 |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )
3		intss1 3839	. 2 |- ( A e. { x e. B | ph } -> |^| { x e. B | ph } C_ A )
4	2, 3	sylbir 134	1 |- ( ( A e. B /\ ps ) -> |^| { x e. B | ph } C_ A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   {crab 2448   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-rab 2453  df-v 2728  df-in 3122  df-ss 3129  df-int 3825

This theorem is referenced by:  onintss  4368  cardonle  7143