Theorem intmin2 3662

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 2620	. . 3 |- { x e. _V | A C_ x } = { x | A C_ x }
2	1	inteqi 3640	. 2 |- |^| { x e. _V | A C_ x } = |^| { x | A C_ x }
3		intmin2.1	. . 3 |- A e. _V
4		intmin 3656	. . 3 |- ( A e. _V -> |^| { x e. _V | A C_ x } = A )
5	3, 4	ax-mp 7	. 2 |- |^| { x e. _V | A C_ x } = A
6	2, 5	eqtr3i 2103	1 |- |^| { x | A C_ x } = A

Syntax hints:   = wceq 1284   e. wcel 1433   {cab 2067   {crab 2352   _Vcvv 2601   C_ wss 2973   |^|cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986  df-int 3637

This theorem is referenced by: (None)