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Theorem elqsn0m 6110
Description: An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)
Assertion
Ref Expression
elqsn0m  dom  R 
/. R
Distinct variable groups:   , R   ,   ,

Proof of Theorem elqsn0m
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqid 2037 . 2 
/. R  /. R
2 eleq2 2098 . . 3  R  R
32exbidv 1703 . 2  R  R
4 eleq2 2098 . . . 4  dom 
R  dom  R
54biimpar 281 . . 3  dom  R  dom  R
6 ecdmn0m 6084 . . 3  dom  R  R
75, 6sylib 127 . 2  dom  R  R
81, 3, 7ectocld 6108 1  dom  R 
/. R
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390   dom cdm 4288  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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  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:  elqsn0  6111  ecelqsdm  6112
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