MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snsssn Structured version   Unicode version

Theorem snsssn 3959
Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
Hypothesis
Ref Expression
sneqr.1  |-  A  e. 
_V
Assertion
Ref Expression
snsssn  |-  ( { A }  C_  { B }  ->  A  =  B )

Proof of Theorem snsssn
StepHypRef Expression
1 sssn 3949 . 2  |-  ( { A }  C_  { B } 
<->  ( { A }  =  (/)  \/  { A }  =  { B } ) )
2 sneqr.1 . . . . . 6  |-  A  e. 
_V
32snnz 3914 . . . . 5  |-  { A }  =/=  (/)
4 df-ne 2600 . . . . 5  |-  ( { A }  =/=  (/)  <->  -.  { A }  =  (/) )
53, 4mpbi 200 . . . 4  |-  -.  { A }  =  (/)
65pm2.21i 125 . . 3  |-  ( { A }  =  (/)  ->  A  =  B )
72sneqr 3958 . . 3  |-  ( { A }  =  { B }  ->  A  =  B )
86, 7jaoi 369 . 2  |-  ( ( { A }  =  (/) 
\/  { A }  =  { B } )  ->  A  =  B )
91, 8sylbi 188 1  |-  ( { A }  C_  { B }  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    C_ wss 3312   (/)c0 3620   {csn 3806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812
  Copyright terms: Public domain W3C validator