Theorem intminss 3953

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 2995	. 2 |- (A e. {x e. B | ph} <-> (A e. B /\ ps))
3		intss1 3942	. 2 |- (A e. {x e. B | ph} -> |^|{x e. B | ph} C_ A)
4	2, 3	sylbir 204	1 |- ((A e. B /\ ps) -> |^|{x e. B | ph} C_ A)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  {crab 2619   C_ wss 3258  |^|cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928

This theorem is referenced by: (None)