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Theorem snec 6103
Description: The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypothesis
Ref Expression
snec.1  _V
Assertion
Ref Expression
snec  { R }  { } /. R

Proof of Theorem snec
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snec.1 . . . 4  _V
2 eceq1 6077 . . . . 5  R  R
32eqeq2d 2048 . . . 4  R  R
41, 3rexsn 3406 . . 3  { }  R  R
54abbii 2150 . 2  {  | 
{ }  R }  {  |  R }
6 df-qs 6048 . 2  { } /. R  {  |  { }  R }
7 df-sn 3373 . 2  { R }  {  |  R }
85, 6, 73eqtr4ri 2068 1  { R }  { } /. R
Colors of variables: wff set class
Syntax hints:   wceq 1242   wcel 1390   {cab 2023  wrex 2301   _Vcvv 2551   {csn 3367  cec 6040   /.cqs 6041
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-bndl 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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-ec 6044  df-qs 6048
This theorem is referenced by: (None)
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