ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ss1o0el1 Unicode version

Theorem ss1o0el1 4209
Description: A subclass of  { (/) } contains the empty set if and only if it equals  { (/) }. (Contributed by BJ and Jim Kingdon, 9-Aug-2024.)
Assertion
Ref Expression
ss1o0el1  |-  ( A 
C_  { (/) }  ->  (
(/)  e.  A  <->  A  =  { (/) } ) )

Proof of Theorem ss1o0el1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex2 2765 . . . 4  |-  ( (/)  e.  A  ->  E. x  x  e.  A )
2 sssnm 3766 . . . 4  |-  ( E. x  x  e.  A  ->  ( A  C_  { (/) }  <-> 
A  =  { (/) } ) )
31, 2syl 14 . . 3  |-  ( (/)  e.  A  ->  ( A 
C_  { (/) }  <->  A  =  { (/) } ) )
43biimpcd 159 . 2  |-  ( A 
C_  { (/) }  ->  (
(/)  e.  A  ->  A  =  { (/) } ) )
5 0ex 4142 . . . 4  |-  (/)  e.  _V
65snid 3635 . . 3  |-  (/)  e.  { (/)
}
7 eleq2 2251 . . 3  |-  ( A  =  { (/) }  ->  (
(/)  e.  A  <->  (/)  e.  { (/)
} ) )
86, 7mpbiri 168 . 2  |-  ( A  =  { (/) }  ->  (/)  e.  A )
94, 8impbid1 142 1  |-  ( A 
C_  { (/) }  ->  (
(/)  e.  A  <->  A  =  { (/) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1363   E.wex 1502    e. wcel 2158    C_ wss 3141   (/)c0 3434   {csn 3604
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-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-nul 4141
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-in 3147  df-ss 3154  df-nul 3435  df-sn 3610
This theorem is referenced by:  exmid01  4210  ss1o0el1o  6925
  Copyright terms: Public domain W3C validator