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

Theorem snssOLD 3720
Description: Obsolete version of snss 3729 as of 1-Jan-2025. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
snssOLD.1  |-  A  e. 
_V
Assertion
Ref Expression
snssOLD  |-  ( A  e.  B  <->  { A }  C_  B )

Proof of Theorem snssOLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 velsn 3611 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21imbi1i 238 . . 3  |-  ( ( x  e.  { A }  ->  x  e.  B
)  <->  ( x  =  A  ->  x  e.  B ) )
32albii 1470 . 2  |-  ( A. x ( x  e. 
{ A }  ->  x  e.  B )  <->  A. x
( x  =  A  ->  x  e.  B
) )
4 dfss2 3146 . 2  |-  ( { A }  C_  B  <->  A. x ( x  e. 
{ A }  ->  x  e.  B ) )
5 snssOLD.1 . . 3  |-  A  e. 
_V
65clel2 2872 . 2  |-  ( A  e.  B  <->  A. x
( x  =  A  ->  x  e.  B
) )
73, 4, 63bitr4ri 213 1  |-  ( A  e.  B  <->  { A }  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351    = wceq 1353    e. wcel 2148   _Vcvv 2739    C_ wss 3131   {csn 3594
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144  df-sn 3600
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator