Theorem intmin2 3797

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 2707	. . 3 |- { x e. _V | A C_ x } = { x | A C_ x }
2	1	inteqi 3775	. 2 |- |^| { x e. _V | A C_ x } = |^| { x | A C_ x }
3		intmin2.1	. . 3 |- A e. _V
4		intmin 3791	. . 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 2162	1 |- |^| { x | A C_ x } = A

Syntax hints:   = wceq 1331   e. wcel 1480   {cab 2125   {crab 2420   _Vcvv 2686   C_ wss 3071   |^|cint 3771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-int 3772

This theorem is referenced by: (None)