ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  intmin2 Unicode version

Theorem intmin2 3641
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
StepHypRef Expression
1 rabab 2575 . . 3  |-  { x  e.  _V  |  A  C_  x }  =  {
x  |  A  C_  x }
21inteqi 3619 . 2  |-  |^| { x  e.  _V  |  A  C_  x }  =  |^| { x  |  A  C_  x }
3 intmin2.1 . . 3  |-  A  e. 
_V
4 intmin 3635 . . 3  |-  ( A  e.  _V  ->  |^| { x  e.  _V  |  A  C_  x }  =  A
)
53, 4ax-mp 7 . 2  |-  |^| { x  e.  _V  |  A  C_  x }  =  A
62, 5eqtr3i 2062 1  |-  |^| { x  |  A  C_  x }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1243    e. wcel 1393   {cab 2026   {crab 2310   _Vcvv 2557    C_ wss 2917   |^|cint 3615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-int 3616
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator