Theorem intmin2 3948

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 2821	. . 3 |- { x e. _V | A C_ x } = { x | A C_ x }
2	1	inteqi 3926	. 2 |- |^| { x e. _V | A C_ x } = |^| { x | A C_ x }
3		intmin2.1	. . 3 |- A e. _V
4		intmin 3942	. . 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 2252	1 |- |^| { x | A C_ x } = A

Syntax hints:   = wceq 1395   e. wcel 2200   {cab 2215   {crab 2512   _Vcvv 2799   C_ wss 3197   |^|cint 3922

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-v 2801  df-in 3203  df-ss 3210  df-int 3923

This theorem is referenced by: (None)