Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  intmin3 Structured version   Unicode version

Theorem intmin3 4070
 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
intmin3.3
Assertion
Ref Expression
intmin3
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem intmin3
StepHypRef Expression
1 intmin3.3 . . 3
2 intmin3.2 . . . 4
32elabg 3075 . . 3
41, 3mpbiri 225 . 2
5 intss1 4057 . 2
64, 5syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wceq 1652   wcel 1725  cab 2421   wss 3312  cint 4042 This theorem is referenced by:  intabs  4353  intid  4413 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-ss 3326  df-int 4043
 Copyright terms: Public domain W3C validator