Theorem intmin2 3954

Description: Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)

Hypothesis

Ref	Expression
intmin2.1	|- A e. _V

Assertion

Ref	Expression
intmin2	|- |^|{x | A C_ x} = A

Distinct variable group:   x,A

Proof of Theorem intmin2

Step	Hyp	Ref	Expression
1		rabab 2877	. . 3 |- {x e. _V | A C_ x} = {x | A C_ x}
2	1	inteqi 3931	. 2 |- |^|{x e. _V | A C_ x} = |^|{x | A C_ x}
3		intmin2.1	. . 3 |- A e. _V
4		intmin 3947	. . 3 |- (A e. _V -> |^|{x e. _V | A C_ x} = A)
5	3, 4	ax-mp 5	. 2 |- |^|{x e. _V | A C_ x} = A
6	2, 5	eqtr3i 2375	1 |- |^|{x | A C_ x} = A

Syntax hints:   = wceq 1642   e. wcel 1710  {cab 2339  {crab 2619  _Vcvv 2860   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-ral 2620  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)