Theorem intmin2 3911

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 2793	. . 3 |- { x e. _V | A C_ x } = { x | A C_ x }
2	1	inteqi 3889	. 2 |- |^| { x e. _V | A C_ x } = |^| { x | A C_ x }
3		intmin2.1	. . 3 |- A e. _V
4		intmin 3905	. . 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 2228	1 |- |^| { x | A C_ x } = A

Syntax hints:   = wceq 1373   e. wcel 2176   {cab 2191   {crab 2488   _Vcvv 2772   C_ wss 3166   |^|cint 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rab 2493  df-v 2774  df-in 3172  df-ss 3179  df-int 3886

This theorem is referenced by: (None)