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Theorem snsssn 3523
Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
Hypothesis
Ref Expression
sneqr.1  _V
Assertion
Ref Expression
snsssn  { }  C_  { }

Proof of Theorem snsssn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfss2 2928 . . 3  { }  C_  { } 
{ }  { }
2 elsn 3382 . . . . 5  { }
3 elsn 3382 . . . . 5  { }
42, 3imbi12i 228 . . . 4  { }  { }
54albii 1356 . . 3  { }  { }
61, 5bitri 173 . 2  { }  C_  { }
7 sneqr.1 . . 3  _V
8 sbceqal 2808 . . 3  _V
97, 8ax-mp 7 . 2
106, 9sylbi 114 1  { }  C_  { }
Colors of variables: wff set class
Syntax hints:   wi 4  wal 1240   wceq 1242   wcel 1390   _Vcvv 2551    C_ wss 2911   {csn 3367
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-in 2918  df-ss 2925  df-sn 3373
This theorem is referenced by: (None)
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