Theorem intmin3 3954

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 2986	. . 3 |- (A e. V -> (A e. {x | ph} <-> ps))
4	1, 3	mpbiri 224	. 2 |- (A e. V -> A e. {x | ph})
5		intss1 3941	. 2 |- (A e. {x | ph} -> |^|{x | ph} C_ A)
6	4, 5	syl 15	1 |- (A e. V -> |^|{x | ph} C_ A)

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   e. wcel 1710  {cab 2339   C_ wss 3257  |^|cint 3926

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927

This theorem is referenced by: (None)