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 2824	. . 3 |- { x e. _V | A C_ x } = { x | A C_ x }
2	1	inteqi 3932	. 2 |- |^| { x e. _V | A C_ x } = |^| { x | A C_ x }
3		intmin2.1	. . 3 |- A e. _V
4		intmin 3948	. . 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 2254	1 |- |^| { x | A C_ x } = A

Syntax hints:   = wceq 1397   e. wcel 2202   {cab 2217   {crab 2514   _Vcvv 2802   C_ wss 3200   |^|cint 3928

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rab 2519  df-v 2804  df-in 3206  df-ss 3213  df-int 3929

This theorem is referenced by: (None)